Millimeter Wave Generation by Relativistic Electron Beams.

DTIC Science & Technology

1984-12-01

frequency and wave vector matching relations for influence of various nonlinear effects on this instability is this four-wave interaction require...following coupled mode equations _ 6 = 6 _ (14)-- v vx (14) ." .’ for the lower hybrid sidebands: v - V 2 - The x component of the resultant vector equation...involves a purely growing modte, a four-wave interaction plitoces is analysed, including a u ap ti wave- vector up-shifted and ilown-shiftes upper

Versatile rogue waves in scalar, vector, and multidimensional nonlinear systems

NASA Astrophysics Data System (ADS)

Chen, Shihua; Baronio, Fabio; Soto-Crespo, Jose M.; Grelu, Philippe; Mihalache, Dumitru

2017-11-01

This review is dedicated to recent progress in the active field of rogue waves, with an emphasis on the analytical prediction of versatile rogue wave structures in scalar, vector, and multidimensional integrable nonlinear systems. We first give a brief outline of the historical background of the rogue wave research, including referring to relevant up-to-date experimental results. Then we present an in-depth discussion of the scalar rogue waves within two different integrable frameworks—the infinite nonlinear Schrödinger (NLS) hierarchy and the general cubic-quintic NLS equation, considering both the self-focusing and self-defocusing Kerr nonlinearities. We highlight the concept of chirped Peregrine solitons, the baseband modulation instability as an origin of rogue waves, and the relation between integrable turbulence and rogue waves, each with illuminating examples confirmed by numerical simulations. Later, we recur to the vector rogue waves in diverse coupled multicomponent systems such as the long-wave short-wave equations, the three-wave resonant interaction equations, and the vector NLS equations (alias Manakov system). In addition to their intriguing bright-dark dynamics, a series of other peculiar structures, such as coexisting rogue waves, watch-hand-like rogue waves, complementary rogue waves, and vector dark three sisters, are reviewed. Finally, for practical considerations, we also remark on higher-dimensional rogue waves occurring in three closely-related (2  +  1)D nonlinear systems, namely, the Davey-Stewartson equation, the composite (2  +  1)D NLS equation, and the Kadomtsev-Petviashvili I equation. As an interesting contrast to the peculiar X-shaped light bullets, a concept of rogue wave bullets intended for high-dimensional systems is particularly put forward by combining contexts in nonlinear optics.

Circularly polarized few-cycle optical rogue waves: rotating reduced Maxwell-Bloch equations.

PubMed

Xu, Shuwei; Porsezian, K; He, Jingsong; Cheng, Yi

2013-12-01

The rotating reduced Maxwell-Bloch (RMB) equations, which describe the propagation of few-cycle optical pulses in a transparent media with two isotropic polarized electronic field components, are derived from a system of complete Maxwell-Bloch equations without using the slowly varying envelope approximations. Two hierarchies of the obtained rational solutions, including rogue waves, which are also called few-cycle optical rogue waves, of the rotating RMB equations are constructed explicitly through degenerate Darboux transformation. In addition to the above, the dynamical evolution of the first-, second-, and third-order few-cycle optical rogue waves are constructed with different patterns. For an electric field E in the three lower-order rogue waves, we find that rogue waves correspond to localized large amplitude oscillations of the polarized electric fields. Further a complementary relationship of two electric field components of rogue waves is discussed in terms of analytical formulas as well as numerical figures.

Multiple and exact soliton solutions of the perturbed Korteweg-de Vries equation of long surface waves in a convective fluid via Painlevé analysis, factorization, and simplest equation methods.

PubMed

Selima, Ehab S; Yao, Xiaohua; Wazwaz, Abdul-Majid

2017-06-01

In this research, the surface waves of a horizontal fluid layer open to air under gravity field and vertical temperature gradient effects are studied. The governing equations of this model are reformulated and converted to a nonlinear evolution equation, the perturbed Korteweg-de Vries (pKdV) equation. We investigate the latter equation, which includes dispersion, diffusion, and instability effects, in order to examine the evolution of long surface waves in a convective fluid. Dispersion relation of the pKdV equation and its properties are discussed. The Painlevé analysis is applied not only to check the integrability of the pKdV equation but also to establish the Bäcklund transformation form. In addition, traveling wave solutions and a general form of the multiple-soliton solutions of the pKdV equation are obtained via Bäcklund transformation, the simplest equation method using Bernoulli, Riccati, and Burgers' equations as simplest equations, and the factorization method.

Analytical and numerical solution for wave reflection from a porous wave absorber

NASA Astrophysics Data System (ADS)

Magdalena, Ikha; Roque, Marian P.

2018-03-01

In this paper, wave reflection from a porous wave absorber is investigated theoretically and numerically. The equations that we used are based on shallow water type model. Modification of motion inside the absorber is by including linearized friction term in momentum equation and introducing a filtered velocity. Here, an analytical solution for wave reflection coefficient from a porous wave absorber over a flat bottom is derived. Numerically, we solve the equations using the finite volume method on a staggered grid. To validate our numerical model, comparison of the numerical reflection coefficient is made against the analytical solution. Further, we implement our numerical scheme to study the evolution of surface waves pass through a porous absorber over varied bottom topography.

Stimulated scattering of electromagnetic waves carrying orbital angular momentum in quantum plasmas.

PubMed

Shukla, P K; Eliasson, B; Stenflo, L

2012-07-01

We investigate stimulated scattering instabilities of coherent circularly polarized electromagnetic (CPEM) waves carrying orbital angular momentum (OAM) in dense quantum plasmas with degenerate electrons and nondegenerate ions. For this purpose, we employ the coupled equations for the CPEM wave vector potential and the driven (by the ponderomotive force of the CPEM waves) equations for the electron and ion plasma oscillations. The electrons are significantly affected by the quantum forces (viz., the quantum statistical pressure, the quantum Bohm potential, as well as the electron exchange and electron correlations due to electron spin), which are included in the framework of the quantum hydrodynamical description of the electrons. Furthermore, our investigation of the stimulated Brillouin instability of coherent CPEM waves uses the generalized ion momentum equation that includes strong ion coupling effects. The nonlinear equations for the coupled CPEM and quantum plasma waves are then analyzed to obtain nonlinear dispersion relations which exhibit stimulated Raman, stimulated Brillouin, and modulational instabilities of CPEM waves carrying OAM. The present results are useful for understanding the origin of scattered light off low-frequency density fluctuations in high-energy density plasmas where quantum effects are eminent.

Local-in-space blow-up criteria for a class of nonlinear dispersive wave equations

NASA Astrophysics Data System (ADS)

Novruzov, Emil

2017-11-01

This paper is concerned with blow-up phenomena for the nonlinear dispersive wave equation on the real line, ut -uxxt +[ f (u) ] x -[ f (u) ] xxx +[ g (u) + f″/(u) 2 ux2 ] x = 0 that includes the Camassa-Holm equation as well as the hyperelastic-rod wave equation (f (u) = ku2 / 2 and g (u) = (3 - k) u2 / 2) as special cases. We establish some a local-in-space blow-up criterion (i.e., a criterion involving only the properties of the data u0 in a neighborhood of a single point) simplifying and precising earlier blow-up criteria for this equation.

Soliton solutions to the fifth-order Korteweg-de Vries equation and their applications to surface and internal water waves

NASA Astrophysics Data System (ADS)

Khusnutdinova, K. R.; Stepanyants, Y. A.; Tranter, M. R.

2018-02-01

We study solitary wave solutions of the fifth-order Korteweg-de Vries equation which contains, besides the traditional quadratic nonlinearity and third-order dispersion, additional terms including cubic nonlinearity and fifth order linear dispersion, as well as two nonlinear dispersive terms. An exact solitary wave solution to this equation is derived, and the dependence of its amplitude, width, and speed on the parameters of the governing equation is studied. It is shown that the derived solution can represent either an embedded or regular soliton depending on the equation parameters. The nonlinear dispersive terms can drastically influence the existence of solitary waves, their nature (regular or embedded), profile, polarity, and stability with respect to small perturbations. We show, in particular, that in some cases embedded solitons can be stable even with respect to interactions with regular solitons. The results obtained are applicable to surface and internal waves in fluids, as well as to waves in other media (plasma, solid waveguides, elastic media with microstructure, etc.).

Dynamic response of a riser under excitation of internal waves

NASA Astrophysics Data System (ADS)

Lou, Min; Yu, Chenglong; Chen, Peng

2015-12-01

In this paper, the dynamic response of a marine riser under excitation of internal waves is studied. With the linear approximation, the governing equation of internal waves is given. Based on the rigid-lid boundary condition assumption, the equation is solved by Thompson-Haskell method. Thus the velocity field of internal waves is obtained by the continuity equation. Combined with the modified Morison formula, using finite element method, the motion equation of riser is solved in time domain with Newmark-β method. The computation programs are compiled to solve the differential equations in time domain. Then we get the numerical results, including riser displacement and transfiguration. It is observed that the internal wave will result in circular shear flow, and the first two modes have a dominant effect on dynamic response of the marine riser. In the high mode, the response diminishes rapidly. In different modes of internal waves, the deformation of riser has different shapes, and the location of maximum displacement shifts. Studies on wave parameters indicate that the wave amplitude plays a considerable role in response displacement of riser, while the wave frequency contributes little. Nevertheless, the internal waves of high wave frequency will lead to a high-frequency oscillation of riser; it possibly gives rise to fatigue crack extension and partial fatigue failure.

Kato Smoothing and Strichartz Estimates for Wave Equations with Magnetic Potentials

NASA Astrophysics Data System (ADS)

D'Ancona, Piero

2015-04-01

Let H be a selfadjoint operator and A a closed operator on a Hilbert space . If A is H-(super)smooth in the sense of Kato-Yajima, we prove that is -(super)smooth. This allows us to include wave and Klein-Gordon equations in the abstract theory at the same level of generality as Schrödinger equations. We give a few applications and in particular, based on the resolvent estimates of Erdogan, Goldberg and Schlag (Forum Mathematicum 21:687-722, 2009), we prove Strichartz estimates for wave equations perturbed with large magnetic potentials on , n ≥ 3.

Finite-amplitude strain waves in laser-excited plates.

PubMed

Mirzade, F Kh

2008-07-09

The governing equations for two-dimensional finite-amplitude longitudinal strain waves in isotropic laser-excited solid plates are derived. Geometric and weak material nonlinearities are included, and the interaction of longitudinal displacements with the field of concentration of non-equilibrium laser-generated atomic defects is taken into account. An asymptotic approach is used to show that the equations are reducible to the Kadomtsev-Petviashvili-Burgers nonlinear evolution equation for a longitudinal self-consistent strain field. It is shown that two-dimensional shock waves can propagate in plates.

Internally electrodynamic particle model: Its experimental basis and its predictions

DOE Office of Scientific and Technical Information (OSTI.GOV)

Zheng-Johansson, J. X., E-mail: jxzj@iofpr.or

2010-03-15

The internally electrodynamic (IED) particle model was derived based on overall experimental observations, with the IED process itself being built directly on three experimental facts: (a) electric charges present with all material particles, (b) an accelerated charge generates electromagnetic waves according to Maxwell's equations and Planck energy equation, and (c) source motion produces Doppler effect. A set of well-known basic particle equations and properties become predictable based on first principles solutions for the IED process; several key solutions achieved are outlined, including the de Broglie phase wave, de Broglie relations, Schroedinger equation, mass, Einstein mass-energy relation, Newton's law of gravity,more » single particle self interference, and electromagnetic radiation and absorption; these equations and properties have long been broadly experimentally validated or demonstrated. A conditioned solution also predicts the Doebner-Goldin equation which emerges to represent a form of long-sought quantum wave equation including gravity. A critical review of the key experiments is given which suggests that the IED process underlies the basic particle equations and properties not just sufficiently but also necessarily.« less

Internally electrodynamic particle model: Its experimental basis and its predictions

NASA Astrophysics Data System (ADS)

Zheng-Johansson, J. X.

2010-03-01

The internally electrodynamic (IED) particle model was derived based on overall experimental observations, with the IED process itself being built directly on three experimental facts: (a) electric charges present with all material particles, (b) an accelerated charge generates electromagnetic waves according to Maxwell’s equations and Planck energy equation, and (c) source motion produces Doppler effect. A set of well-known basic particle equations and properties become predictable based on first principles solutions for the IED process; several key solutions achieved are outlined, including the de Broglie phase wave, de Broglie relations, Schrödinger equation, mass, Einstein mass-energy relation, Newton’s law of gravity, single particle self interference, and electromagnetic radiation and absorption; these equations and properties have long been broadly experimentally validated or demonstrated. A conditioned solution also predicts the Doebner-Goldin equation which emerges to represent a form of long-sought quantum wave equation including gravity. A critical review of the key experiments is given which suggests that the IED process underlies the basic particle equations and properties not just sufficiently but also necessarily.

Modeling RF Fields in Hot Plasmas with Parallel Full Wave Code

NASA Astrophysics Data System (ADS)

Spencer, Andrew; Svidzinski, Vladimir; Zhao, Liangji; Galkin, Sergei; Kim, Jin-Soo

2016-10-01

FAR-TECH, Inc. is developing a suite of full wave RF plasma codes. It is based on a meshless formulation in configuration space with adapted cloud of computational points (CCP) capability and using the hot plasma conductivity kernel to model the nonlocal plasma dielectric response. The conductivity kernel is calculated by numerically integrating the linearized Vlasov equation along unperturbed particle trajectories. Work has been done on the following calculations: 1) the conductivity kernel in hot plasmas, 2) a monitor function based on analytic solutions of the cold-plasma dispersion relation, 3) an adaptive CCP based on the monitor function, 4) stencils to approximate the wave equations on the CCP, 5) the solution to the full wave equations in the cold-plasma model in tokamak geometry for ECRH and ICRH range of frequencies, and 6) the solution to the wave equations using the calculated hot plasma conductivity kernel. We will present results on using a meshless formulation on adaptive CCP to solve the wave equations and on implementing the non-local hot plasma dielectric response to the wave equations. The presentation will include numerical results of wave propagation and absorption in the cold and hot tokamak plasma RF models, using DIII-D geometry and plasma parameters. Work is supported by the U.S. DOE SBIR program.

Iterative Methods to Solve Linear RF Fields in Hot Plasma

NASA Astrophysics Data System (ADS)

Spencer, Joseph; Svidzinski, Vladimir; Evstatiev, Evstati; Galkin, Sergei; Kim, Jin-Soo

2014-10-01

Most magnetic plasma confinement devices use radio frequency (RF) waves for current drive and/or heating. Numerical modeling of RF fields is an important part of performance analysis of such devices and a predictive tool aiding design and development of future devices. Prior attempts at this modeling have mostly used direct solvers to solve the formulated linear equations. Full wave modeling of RF fields in hot plasma with 3D nonuniformities is mostly prohibited, with memory demands of a direct solver placing a significant limitation on spatial resolution. Iterative methods can significantly increase spatial resolution. We explore the feasibility of using iterative methods in 3D full wave modeling. The linear wave equation is formulated using two approaches: for cold plasmas the local cold plasma dielectric tensor is used (resolving resonances by particle collisions), while for hot plasmas the conductivity kernel (which includes a nonlocal dielectric response) is calculated by integrating along test particle orbits. The wave equation is discretized using a finite difference approach. The initial guess is important in iterative methods, and we examine different initial guesses including the solution to the cold plasma wave equation. Work is supported by the U.S. DOE SBIR program.

Matter rogue waves for the three-component Gross-Pitaevskii equations in the spinor Bose-Einstein condensates.

PubMed

Sun, Wen-Rong; Wang, Lei

2018-01-01

To show the existence and properties of matter rogue waves in an F =1 spinor Bose-Einstein condensate (BEC), we work on the three-component Gross-Pitaevskii (GP) equations. Via the Darboux-dressing transformation, we obtain a family of rational solutions describing the extreme events, i.e. rogue waves. This family of solutions includes bright-dark-bright and bright-bright-bright rogue waves. The algebraic construction depends on Lax matrices and their Jordan form. The conditions for the existence of rogue wave solutions in an F =1 spinor BEC are discussed. For the three-component GP equations, if there is modulation instability, it is of baseband type only, confirming our analytic conditions. The energy transfers between the waves are discussed.

Matter rogue waves for the three-component Gross-Pitaevskii equations in the spinor Bose-Einstein condensates

NASA Astrophysics Data System (ADS)

Sun, Wen-Rong; Wang, Lei

2018-01-01

To show the existence and properties of matter rogue waves in an F=1 spinor Bose-Einstein condensate (BEC), we work on the three-component Gross-Pitaevskii (GP) equations. Via the Darboux-dressing transformation, we obtain a family of rational solutions describing the extreme events, i.e. rogue waves. This family of solutions includes bright-dark-bright and bright-bright-bright rogue waves. The algebraic construction depends on Lax matrices and their Jordan form. The conditions for the existence of rogue wave solutions in an F=1 spinor BEC are discussed. For the three-component GP equations, if there is modulation instability, it is of baseband type only, confirming our analytic conditions. The energy transfers between the waves are discussed.

A Note on the Wave Action Density of a Viscous Instability Mode on a Laminar Free-shear Flow

NASA Technical Reports Server (NTRS)

Balsa, Thomas F.

1994-01-01

Using the assumptions of an incompressible and viscous flow at large Reynolds number, we derive the evolution equation for the wave action density of an instability wave traveling on top of a laminar free-shear flow. The instability is considered to be viscous; the purpose of the present work is to include the cumulative effect of the (locally) small viscous correction to the wave, over length and time scales on which the underlying base flow appears inhomogeneous owing to its viscous diffusion. As such, we generalize our previous work for inviscid waves. This generalization appears as an additional (but usually non-negligible) term in the equation for the wave action. The basic structure of the equation remains unaltered.

Uniform high order spectral methods for one and two dimensional Euler equations

NASA Technical Reports Server (NTRS)

Cai, Wei; Shu, Chi-Wang

1991-01-01

Uniform high order spectral methods to solve multi-dimensional Euler equations for gas dynamics are discussed. Uniform high order spectral approximations with spectral accuracy in smooth regions of solutions are constructed by introducing the idea of the Essentially Non-Oscillatory (ENO) polynomial interpolations into the spectral methods. The authors present numerical results for the inviscid Burgers' equation, and for the one dimensional Euler equations including the interactions between a shock wave and density disturbance, Sod's and Lax's shock tube problems, and the blast wave problem. The interaction between a Mach 3 two dimensional shock wave and a rotating vortex is simulated.

Numerical Modeling of Electroacoustic Logging Including Joule Heating

NASA Astrophysics Data System (ADS)

Plyushchenkov, Boris D.; Nikitin, Anatoly A.; Turchaninov, Victor I.

It is well known that electromagnetic field excites acoustic wave in a porous elastic medium saturated with fluid electrolyte due to electrokinetic conversion effect. Pride's equations describing this process are written in isothermal approximation. Update of these equations, which allows to take influence of Joule heating on acoustic waves propagation into account, is proposed here. This update includes terms describing the initiation of additional acoustic waves excited by thermoelastic stresses and the heat conduction equation with right side defined by Joule heating. Results of numerical modeling of several problems of propagation of acoustic waves excited by an electric field source with and without consideration of Joule heating effect in their statements are presented. From these results, it follows that influence of Joule heating should be taken into account at the numerical simulation of electroacoustic logging and at the interpretation of its log data.

Numerical solution of equations governing longitudinal suspension line wave motion during the parachute unfurling process. Ph.D. Thesis - George Washington Univ., Washington, D. C.

NASA Technical Reports Server (NTRS)

Poole, L. R.

1973-01-01

Equations are presented which govern the dynamics of the lines-first parachute unfurling process, including wave motion in the parachute suspension lines. Techniques are developed for obtaining numerical solutions to the governing equations. Histories of tension at test data, and generally good agreement is observed. Errors in computed results are attributed to several areas of uncertainty, the most significant being a poorly defined boundary condition on the wave motion at the vehicle-suspension line boundary.

Nonlinear Waves and Inverse Scattering

DTIC Science & Technology

1989-01-01

transform provides a linearization.’ Well known systems include the Kadomtsev - Petviashvili , Davey-Stewartson and Self-Dual Yang-Mills equations . The d...which employs inverse scattering theory in order to linearize the given nonlinear equation . I.S.T. has led to new developments in both fields: inverse...scattering and nonlinear wave equations . Listed below are some of the problems studied and a short description of results. - Multidimensional

Localized light waves: Paraxial and exact solutions of the wave equation (a review)

NASA Astrophysics Data System (ADS)

Kiselev, A. P.

2007-04-01

Simple explicit localized solutions are systematized over the whole space of a linear wave equation, which models the propagation of optical radiation in a linear approximation. Much attention has been paid to exact solutions (which date back to the Bateman findings) that describe wave beams (including Bessel-Gauss beams) and wave packets with a Gaussian localization with respect to the spatial variables and time. Their asymptotics with respect to free parameters and at large distances are presented. A similarity between these exact solutions and harmonic in time fields obtained in the paraxial approximation based on the Leontovich-Fock parabolic equation has been studied. Higher-order modes are considered systematically using the separation of variables method. The application of the Bateman solutions of the wave equation to the construction of solutions to equations with dispersion and nonlinearity and their use in wavelet analysis, as well as the summation of Gaussian beams, are discussed. In addition, solutions localized at infinity known as the Moses-Prosser “acoustic bullets”, as well as their harmonic in time counterparts, “ X waves”, waves from complex sources, etc., have been considered. Everywhere possible, the most elementary mathematical formalism is used.

Generalized wave operators, weighted Killing fields, and perturbations of higher dimensional spacetimes

NASA Astrophysics Data System (ADS)

Araneda, Bernardo

2018-04-01

We present weighted covariant derivatives and wave operators for perturbations of certain algebraically special Einstein spacetimes in arbitrary dimensions, under which the Teukolsky and related equations become weighted wave equations. We show that the higher dimensional generalization of the principal null directions are weighted conformal Killing vectors with respect to the modified covariant derivative. We also introduce a modified Laplace–de Rham-like operator acting on tensor-valued differential forms, and show that the wave-like equations are, at the linear level, appropriate projections off shell of this operator acting on the curvature tensor; the projection tensors being made out of weighted conformal Killing–Yano tensors. We give off shell operator identities that map the Einstein and Maxwell equations into weighted scalar equations, and using adjoint operators we construct solutions of the original field equations in a compact form from solutions of the wave-like equations. We study the extreme and zero boost weight cases; extreme boost corresponding to perturbations of Kundt spacetimes (which includes near horizon geometries of extreme black holes), and zero boost to static black holes in arbitrary dimensions. In 4D our results apply to Einstein spacetimes of Petrov type D and make use of weighted Killing spinors.

A generalized simplest equation method and its application to the Boussinesq-Burgers equation.

PubMed

Sudao, Bilige; Wang, Xiaomin

2015-01-01

In this paper, a generalized simplest equation method is proposed to seek exact solutions of nonlinear evolution equations (NLEEs). In the method, we chose a solution expression with a variable coefficient and a variable coefficient ordinary differential auxiliary equation. This method can yield a Bäcklund transformation between NLEEs and a related constraint equation. By dealing with the constraint equation, we can derive infinite number of exact solutions for NLEEs. These solutions include the traveling wave solutions, non-traveling wave solutions, multi-soliton solutions, rational solutions, and other types of solutions. As applications, we obtained wide classes of exact solutions for the Boussinesq-Burgers equation by using the generalized simplest equation method.

A Generalized Simplest Equation Method and Its Application to the Boussinesq-Burgers Equation

PubMed Central

Sudao, Bilige; Wang, Xiaomin

2015-01-01

In this paper, a generalized simplest equation method is proposed to seek exact solutions of nonlinear evolution equations (NLEEs). In the method, we chose a solution expression with a variable coefficient and a variable coefficient ordinary differential auxiliary equation. This method can yield a Bäcklund transformation between NLEEs and a related constraint equation. By dealing with the constraint equation, we can derive infinite number of exact solutions for NLEEs. These solutions include the traveling wave solutions, non-traveling wave solutions, multi-soliton solutions, rational solutions, and other types of solutions. As applications, we obtained wide classes of exact solutions for the Boussinesq-Burgers equation by using the generalized simplest equation method. PMID:25973605

A three-dimensional, finite element model for coastal and estuarine circulation

USGS Publications Warehouse

Walters, R.A.

1992-01-01

This paper describes the development and application of a three-dimensional model for coastal and estuarine circulation. The model uses a harmonic expansion in time and a finite element discretization in space. All nonlinear terms are retained, including quadratic bottom stress, advection and wave transport (continuity nonlinearity). The equations are solved as a global and a local problem, where the global problem is the solution of the wave equation formulation of the shallow water equations, and the local problem is the solution of the momentum equation for the vertical velocity profile. These equations are coupled to the advection-diffusion equation for salt so that density gradient forcing is included in the momentum equations. The model is applied to a study of Delaware Bay, U.S.A., where salinity intrusion is the primary focus. ?? 1991.

Exact soliton of (2 + 1)-dimensional fractional Schrödinger equation

NASA Astrophysics Data System (ADS)

Rizvi, S. T. R.; Ali, K.; Bashir, S.; Younis, M.; Ashraf, R.; Ahmad, M. O.

2017-07-01

The nonlinear fractional Schrödinger equation is the basic equation of fractional quantum mechanics introduced by Nick Laskin in 2002. We apply three tools to solve this mathematical-physical model. First, we find the solitary wave solutions including the trigonometric traveling wave solutions, bell and kink shape solitons using the F-expansion and Improve F-expansion method. We also obtain the soliton solution, singular soliton solutions, rational function solution and elliptic integral function solutions, with the help of the extended trial equation method.

A comparative study of diffraction of shallow-water waves by high-level IGN and GN equations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Zhao, B.B.; Ertekin, R.C.; College of Shipbuilding Engineering, Harbin Engineering University, 150001 Harbin

2015-02-15

This work is on the nonlinear diffraction analysis of shallow-water waves, impinging on submerged obstacles, by two related theories, namely the classical Green–Naghdi (GN) equations and the Irrotational Green–Naghdi (IGN) equations, both sets of equations being at high levels and derived for incompressible and inviscid flows. Recently, the high-level Green–Naghdi equations have been applied to some wave transformation problems. The high-level IGN equations have also been used in the last decade to study certain wave propagation problems. However, past works on these theories used different numerical methods to solve these nonlinear and unsteady sets of differential equations and at differentmore » levels. Moreover, different physical problems have been solved in the past. Therefore, it has not been possible to understand the differences produced by these two sets of theories and their range of applicability so far. We are thus motivated to make a direct comparison of the results produced by these theories by use of the same numerical method to solve physically the same wave diffraction problems. We focus on comparing these two theories by using similar codes; only the equations used are different but other parts of the codes, such as the wave-maker, damping zone, discretion method, matrix solver, etc., are exactly the same. This way, we eliminate many potential sources of differences that could be produced by the solution of different equations. The physical problems include the presence of various submerged obstacles that can be used for example as breakwaters or to represent the continental shelf. A numerical wave tank is created by placing a wavemaker on one end and a wave absorbing beach on the other. The nonlinear and unsteady sets of differential equations are solved by the finite-difference method. The results are compared with different equations as well as with the available experimental data.« less

A comparative study of diffraction of shallow-water waves by high-level IGN and GN equations

NASA Astrophysics Data System (ADS)

Zhao, B. B.; Ertekin, R. C.; Duan, W. Y.

2015-02-01

This work is on the nonlinear diffraction analysis of shallow-water waves, impinging on submerged obstacles, by two related theories, namely the classical Green-Naghdi (GN) equations and the Irrotational Green-Naghdi (IGN) equations, both sets of equations being at high levels and derived for incompressible and inviscid flows. Recently, the high-level Green-Naghdi equations have been applied to some wave transformation problems. The high-level IGN equations have also been used in the last decade to study certain wave propagation problems. However, past works on these theories used different numerical methods to solve these nonlinear and unsteady sets of differential equations and at different levels. Moreover, different physical problems have been solved in the past. Therefore, it has not been possible to understand the differences produced by these two sets of theories and their range of applicability so far. We are thus motivated to make a direct comparison of the results produced by these theories by use of the same numerical method to solve physically the same wave diffraction problems. We focus on comparing these two theories by using similar codes; only the equations used are different but other parts of the codes, such as the wave-maker, damping zone, discretion method, matrix solver, etc., are exactly the same. This way, we eliminate many potential sources of differences that could be produced by the solution of different equations. The physical problems include the presence of various submerged obstacles that can be used for example as breakwaters or to represent the continental shelf. A numerical wave tank is created by placing a wavemaker on one end and a wave absorbing beach on the other. The nonlinear and unsteady sets of differential equations are solved by the finite-difference method. The results are compared with different equations as well as with the available experimental data.

Wave-packet continuum-discretization approach to ion-atom collisions including rearrangement: Application to differential ionization in proton-hydrogen scattering

NASA Astrophysics Data System (ADS)

Abdurakhmanov, I. B.; Bailey, J. J.; Kadyrov, A. S.; Bray, I.

2018-03-01

In this work, we develop a wave-packet continuum-discretization approach to ion-atom collisions that includes rearrangement processes. The total scattering wave function is expanded using a two-center basis built from wave-packet pseudostates. The exact three-body Schrödinger equation is converted into coupled-channel differential equations for time-dependent expansion coefficients. In the asymptotic region these time-dependent coefficients represent transition amplitudes for all processes including elastic scattering, excitation, ionization, and electron capture. The wave-packet continuum-discretization approach is ideal for differential ionization studies as it allows one to generate pseudostates with arbitrary energies and distribution. The approach is used to calculate the double differential cross section for ionization in proton collisions with atomic hydrogen. Overall good agreement with experiment is obtained for all considered cases.

Exact solitary wave solution for higher order nonlinear Schrodinger equation using He's variational iteration method

NASA Astrophysics Data System (ADS)

Rani, Monika; Bhatti, Harbax S.; Singh, Vikramjeet

2017-11-01

In optical communication, the behavior of the ultrashort pulses of optical solitons can be described through nonlinear Schrodinger equation. This partial differential equation is widely used to contemplate a number of physically important phenomena, including optical shock waves, laser and plasma physics, quantum mechanics, elastic media, etc. The exact analytical solution of (1+n)-dimensional higher order nonlinear Schrodinger equation by He's variational iteration method has been presented. Our proposed solutions are very helpful in studying the solitary wave phenomena and ensure rapid convergent series and avoid round off errors. Different examples with graphical representations have been given to justify the capability of the method.

Two types of nonlinear wave equations for diffractive beams in bubbly liquids with nonuniform bubble number density.

PubMed

Kanagawa, Tetsuya

2015-05-01

This paper theoretically treats the weakly nonlinear propagation of diffracted sound beams in nonuniform bubbly liquids. The spatial distribution of the number density of the bubbles, initially in a quiescent state, is assumed to be a slowly varying function of the spatial coordinates; the amplitude of variation is assumed to be small compared to the mean number density. A previous derivation method of nonlinear wave equations for plane progressive waves in uniform bubbly liquids [Kanagawa, Yano, Watanabe, and Fujikawa (2010). J. Fluid Sci. Technol. 5(3), 351-369] is extended to handle quasi-plane beams in weakly nonuniform bubbly liquids. The diffraction effect is incorporated by adding a relation that scales the circular sound source diameter to the wavelength into the original set of scaling relations composed of nondimensional physical parameters. A set of basic equations for bubbly flows is composed of the averaged equations of mass and momentum, the Keller equation for bubble wall, and supplementary equations. As a result, two types of evolution equations, a nonlinear Schrödinger equation including dissipation, diffraction, and nonuniform effects for high-frequency short-wavelength case, and a Khokhlov-Zabolotskaya-Kuznetsov equation including dispersion and nonuniform effects for low-frequency long-wavelength case, are derived from the basic set.

Dynamics of nonautonomous discrete rogue wave solutions for an Ablowitz-Musslimani equation with PT-symmetric potential.

PubMed

Yu, Fajun

2017-02-01

Starting from a discrete spectral problem, we derive a hierarchy of nonlinear discrete equations which include the Ablowitz-Ladik (AL) equation. We analytically study the discrete rogue-wave (DRW) solutions of AL equation with three free parameters. The trajectories of peaks and depressions of profiles for the first- and second-order DRWs are produced by means of analytical and numerical methods. In particular, we study the solutions with dispersion in parity-time ( PT) symmetric potential for Ablowitz-Musslimani equation. And we consider the non-autonomous DRW solutions, parameters controlling and their interactions with variable coefficients, and predict the long-living rogue wave solutions. Our results might provide useful information for potential applications of synthetic PT symmetric systems in nonlinear optics and condensed matter physics.

New compacton soliton solutions and solitary patterns solutions of nonlinearly dispersive Boussinesq equations

NASA Astrophysics Data System (ADS)

Yan, Zhenya; Bluman, George

2002-11-01

The special exact solutions of nonlinearly dispersive Boussinesq equations (called B( m, n) equations), utt- uxx- a( un) xx+ b( um) xxxx=0, is investigated by using four direct ansatze. As a result, abundant new compactons: solitons with the absence of infinite wings, solitary patterns solutions having infinite slopes or cups, solitary waves and singular periodic wave solutions of these two equations are obtained. The variant is extended to include linear dispersion to support compactons and solitary patterns in the linearly dispersive Boussinesq equations with m=1. Moreover, another new compacton solution of the special case, B(2,2) equation, is also found.

CMS-Wave

DTIC Science & Technology

2015-10-30

Coastal Inlets Research Program CMS -Wave CMS -Wave is a two-dimensional spectral wind-wave generation and transformation model that employs a forward...marching, finite-difference method to solve the wave action conservation equation. Capabilities of CMS -Wave include wave shoaling, refraction... CMS -Wave can be used in either on a half- or full-plane mode, with primary waves propagating from the seaward boundary toward shore. It can

Alfvén wave interactions in the solar wind

NASA Astrophysics Data System (ADS)

Webb, G. M.; McKenzie, J. F.; Hu, Q.; le Roux, J. A.; Zank, G. P.

2012-11-01

Alfvén wave mixing (interaction) equations used in locally incompressible turbulence transport equations in the solar wind are analyzed from the perspective of linear wave theory. The connection between the wave mixing equations and non-WKB Alfven wave driven wind theories are delineated. We discuss the physical wave energy equation and the canonical wave energy equation for non-WKB Alfven waves and the WKB limit. Variational principles and conservation laws for the linear wave mixing equations for the Heinemann and Olbert non-WKB wind model are obtained. The connection with wave mixing equations used in locally incompressible turbulence transport in the solar wind are discussed.

Optimization of one-way wave equations.

USGS Publications Warehouse

Lee, M.W.; Suh, S.Y.

1985-01-01

The theory of wave extrapolation is based on the square-root equation or one-way equation. The full wave equation represents waves which propagate in both directions. On the contrary, the square-root equation represents waves propagating in one direction only. A new optimization method presented here improves the dispersion relation of the one-way wave equation. -from Authors

Asymptotic analysis of dissipative waves with applications to their numerical simulation

NASA Technical Reports Server (NTRS)

Hagstrom, Thomas

1990-01-01

Various problems involving the interplay of asymptotics and numerics in the analysis of wave propagation in dissipative systems are studied. A general approach to the asymptotic analysis of linear, dissipative waves is developed. It was applied to the derivation of asymptotic boundary conditions for numerical solutions on unbounded domains. Applications include the Navier-Stokes equations. Multidimensional traveling wave solutions to reaction-diffusion equations are also considered. A preliminary numerical investigation of a thermo-diffusive model of flame propagation in a channel with heat loss at the walls is presented.

Parabola solitons for the nonautonomous KP equation in fluids and plasmas

DOE Office of Scientific and Technical Information (OSTI.GOV)

Yu, Xin, E-mail: yuxin@buaa.edu.cn; Sun, Zhi-Yuan

Under investigation in this paper is a nonautonomous Kadomtsev–Petviashvili (KP) equation in fluids and plasmas. The integrability of this equation is examined via the Painlevé analysis and its multi-soliton solutions are constructed. A constraint is proposed to ensure the existence of parabola solitons for such KP equation. Based on the constructed solutions, the solitonic propagation and interaction, including the elastic interaction, inelastic interaction and soliton resonance for parabola solitons, are discussed. The results might be useful for shallow water wave and rogue wave.

Parabola solitons for the nonautonomous KP equation in fluids and plasmas

NASA Astrophysics Data System (ADS)

Yu, Xin; Sun, Zhi-Yuan

2016-04-01

Under investigation in this paper is a nonautonomous Kadomtsev-Petviashvili (KP) equation in fluids and plasmas. The integrability of this equation is examined via the Painlevé analysis and its multi-soliton solutions are constructed. A constraint is proposed to ensure the existence of parabola solitons for such KP equation. Based on the constructed solutions, the solitonic propagation and interaction, including the elastic interaction, inelastic interaction and soliton resonance for parabola solitons, are discussed. The results might be useful for shallow water wave and rogue wave.

A note on specific variability of long surface gravity waves and drag coefficient in coastal upwelling zone

NASA Astrophysics Data System (ADS)

Krzyścin, Janusz

1990-01-01

In this paper we solve analytically wave kinematic equations and the wave energy transport equation, for basic long surface gravity wave in the coastal upwelling zone. Using Gent and Taylor's (1978) parameterization of drag coefficient (which includes interaction between long surface waves and the air flow) we find variability of this coefficient due to wave amplification and refraction caused by specific surface water current in the region. The drag coefficient grows towards the shore. The growth is faster for stronger current. When the angle between waves and the current is less than 90° the growth is mainly connected with the waves steepness, but when the angle is larger, it is caused by relative growth of the wave phase velocity.

An exact solution to the relativistic equation of motion of a charged particle driven by a linearly polarized electromagnetic wave

NASA Technical Reports Server (NTRS)

Shebalin, John V.

1988-01-01

An exact analytic solution is found for a basic electromagnetic wave-charged particle interaction by solving the nonlinear equations of motion. The particle position, velocity, and corresponding time are found to be explicit functions of the total phase of the wave. Particle position and velocity are thus implicit functions of time. Applications include describing the motion of a free electron driven by an intense laser beam..

Nonlinear Ocean Waves

DTIC Science & Technology

1994-01-06

for all of this work is the fact that the Kadomtsev - Petviashvili equation , a1(atu + ui)xU + a.3u) + ay2u = 0, (KP) describes approximately the evolution...the contents of these two papers. (a) Numerically induced chaos The cubic-nonlinear Schrtdinger equation in one dimension, iatA +,2V + 21i,1 =0, (NLS...arises in several physical contexts, including the evolution of nearly monochromatic, one-dimensional waves in deep water. The equation is known to be

High-order rogue waves of the Benjamin-Ono equation and the nonlocal nonlinear Schrödinger equation

NASA Astrophysics Data System (ADS)

Liu, Wei

2017-10-01

High-order rogue wave solutions of the Benjamin-Ono equation and the nonlocal nonlinear Schrödinger equation are derived by employing the bilinear method, which are expressed by simple polynomials. Typical dynamics of these high-order rogue waves are studied by analytical and graphical ways. For the Benjamin-Ono equation, there are two types of rogue waves, namely, bright rogue waves and dark rogue waves. In particular, the fundamental rogue wave pattern is different from the usual fundamental rogue wave patterns in other soliton equations. For the nonlocal nonlinear Schrödinger equation, the exact explicit rogue wave solutions up to the second order are presented. Typical rogue wave patterns such as Peregrine-type, triple and fundamental rogue waves are put forward. These high-order rogue wave patterns have not been shown before in the nonlocal Schrödinger equation.

Fluid equations with nonlinear wave-particle resonances^

NASA Astrophysics Data System (ADS)

Mattor, Nathan

1997-11-01

We have derived fluid equations that include linear and nonlinear wave-particle resonance effects. This greatly extends previous ``Landau-fluid'' closures, which include linear Landau damping. (G.W. Hammett and F.W. Perkins, Phys. Rev. Lett. 64,) 3019 (1990).^, (Z. Chang and J. D. Callen, Phys. Fluids B 4,) 1167 (1992). The new fluid equations are derived with no approximation regarding nonlinear kinetic interaction, and so additionally include numerous nonlinear kinetic effects. The derivation starts with the electrostatic drift kinetic equation for simplicity, with a Maxwellian distribution function. Fluid closure is accomplished through a simple integration trick applied to the drift kinetic equation, using the property that the nth moment of Maxwellian distribution is related to the nth derivative. The result is a compact closure term appearing in the highest moment equation, a term which involves a plasma dispersion function of the electrostatic field and its derivatives. The new term reduces to the linear closures in appropriate limits, so both approaches retain linear Landau damping. But the nonlinearly closed equations have additional desirable properties. Unlike linear closures, the nonlinear closure retains the time-reversibility of the original kinetic equation. We have shown directly that the nonlinear closure retains at least two nonlinear resonance effects: wave-particle trapping and Compton scattering. Other nonlinear kinetic effects are currently under investigation. The new equations correct two previous discrepancies between kinetic and Landau-fluid predictions, including a propagator discrepancy (N. Mattor, Phys. Fluids B 4,) 3952 (1992). and a numerical discrepancy for the 3-mode shearless bounded slab ITG problem. (S. E. Parker et al.), Phys. Plasmas 1, 1461 (1994). ^* In collaboration with S. E. Parker, Department of Physics, University of Colorado, Boulder. ^ Work performed at LLNL under DoE contract No. W7405-ENG-48.

Controllable rogue waves in the nonautonomous nonlinear system with a linear potential

NASA Astrophysics Data System (ADS)

Dai, C. Q.; Zheng, C. L.; Zhu, H. P.

2012-04-01

Based on the similarity transformation connected the nonautonomous nonlinear Schrödinger equation with the autonomous nonlinear Schrödinger equation, we firstly derive self-similar rogue wave solutions (rational solutions) for the nonautonomous nonlinear system with a linear potential. Then, we investigate the controllable behaviors of one-rogue wave, two-rogue wave and rogue wave triplets in a soliton control system. Our results demonstrate that the propagation behaviors of rogue waves, including postpone, sustainment, recurrence and annihilation, can be manipulated by choosing the relation between the maximum value of the effective propagation distance Z m and the parameter Z 0. Moreover, the excitation time of controllable rogue waves is decided by the parameter T 0.

Survey of Coherent Approximately 1 Hz Waves in Mercury's Inner Magnetosphere from MESSENGER Observations

NASA Technical Reports Server (NTRS)

Boardsen, Scott A.; Slavin, James A.; Anderson, Brian J.; Korth, Haje; Schriver, David; Solomon, Sean C.

2012-01-01

We summarize observations by the MESSENGER spacecraft of highly coherent waves at frequencies between 0.4 and 5 Hz in Mercury's inner magnetosphere. This survey covers the time period from 24 March to 25 September 2011, or 2.1 Mercury years. These waves typically exhibit banded harmonic structure that drifts in frequency as the spacecraft traverses the magnetic equator. The waves are seen at all magnetic local times, but their observed rate of occurrence is much less on the dayside, at least in part the result of MESSENGER's orbit. On the nightside, on average, wave power is maximum near the equator and decreases with increasing magnetic latitude, consistent with an equatorial source. When the spacecraft traverses the plasma sheet during its equatorial crossings, wave power is a factor of 2 larger than for equatorial crossings that do not cross the plasma sheet. The waves are highly transverse at large magnetic latitudes but are more compressional near the equator. However, at the equator the transverse component of these waves increases relative to the compressional component as the degree of polarization decreases. Also, there is a substantial minority of events that are transverse at all magnetic latitudes, including the equator. A few of these latter events could be interpreted as ion cyclotron waves. In general, the waves tend to be strongly linear and characterized by values of the ellipticity less than 0.3 and wave-normal angles peaked near 90 deg. Their maxima in wave power at the equator coupled with their narrow-band character suggests that these waves might be generated locally in loss cone plasma characterized by high values of the ratio beta of plasma pressure to magnetic pressure. Presumably both electromagnetic ion cyclotron waves and electromagnetic ion Bernstein waves can be generated by ion loss cone distributions. If proton beta decreases with increasing magnetic latitude along a field line, then electromagnetic ion Bernstein waves are predicted to transition from compressional to transverse, a pattern consistent with our observations. We hypothesize that these local instabilities can lead to enhanced ion precipitation and directly feed field-line resonances.

Existence and Stability of Spatial Plane Waves for the Incompressible Navier-Stokes in R^3

NASA Astrophysics Data System (ADS)

Correia, Simão; Figueira, Mário

2018-03-01

We consider the three-dimensional incompressible Navier-Stokes equation on the whole space. We observe that this system admits a L^∞ family of global spatial plane wave solutions, which are connected with the two-dimensional equation. We then proceed to prove local well-posedness over a space which includes L^3(R^3) and these solutions. Finally, we prove L^3-stability of spatial plane waves, with no condition on their size.

Exact Solutions for the Integrable Sixth-Order Drinfeld-Sokolov-Satsuma-Hirota System by the Analytical Methods.

PubMed

Manafian Heris, Jalil; Lakestani, Mehrdad

2014-01-01

We establish exact solutions including periodic wave and solitary wave solutions for the integrable sixth-order Drinfeld-Sokolov-Satsuma-Hirota system. We employ this system by using a generalized (G'/G)-expansion and the generalized tanh-coth methods. These methods are developed for searching exact travelling wave solutions of nonlinear partial differential equations. It is shown that these methods, with the help of symbolic computation, provide a straightforward and powerful mathematical tool for solving nonlinear partial differential equations.

Wave propagation in a quasi-chemical equilibrium plasma

NASA Technical Reports Server (NTRS)

Fang, T.-M.; Baum, H. R.

1975-01-01

Wave propagation in a quasi-chemical equilibrium plasma is studied. The plasma is infinite and without external fields. The chemical reactions are assumed to result from the ionization and recombination processes. When the gas is near equilibrium, the dominant role describing the evolution of a reacting plasma is played by the global conservation equations. These equations are first derived and then used to study the small amplitude wave motion for a near-equilibrium situation. Nontrivial damping effects have been obtained by including the conduction current terms.

Exact travelling wave solutions for a diffusion-convection equation in two and three spatial dimensions

NASA Astrophysics Data System (ADS)

Elwakil, S. A.; El-Labany, S. K.; Zahran, M. A.; Sabry, R.

2004-04-01

The modified extended tanh-function method were applied to the general class of nonlinear diffusion-convection equations where the concentration-dependent diffusivity, D( u), was taken to be a constant while the concentration-dependent hydraulic conductivity, K( u) were taken to be in a power law. The obtained solutions include rational-type, triangular-type, singular-type, and solitary wave solutions. In fact, the profile of the obtained solitary wave solutions resemble the characteristics of a shock-wave like structure for an arbitrary m (where m>1 is the power of the nonlinear convection term).

On traveling waves in beams

NASA Technical Reports Server (NTRS)

Leonard, Robert W; Budiansky, Bernard

1954-01-01

The basic equations of Timoshenko for the motion of vibrating nonuniform beams, which allow for effects of transverse shear deformation and rotary inertia, are presented in several forms, including one in which the equations are written in the directions of the characteristics. The propagation of discontinuities in moment and shear, as governed by these equations, is discussed. Numerical traveling-wave solutions are obtained for some elementary problems of finite uniform beams for which the propagation velocities of bending and shear discontinuities are taken to be equal. These solutions are compared with modal solutions of Timoshenko's equations and, in some cases, with exact closed solutions. (author)

General high-order breathers and rogue waves in the (3 + 1) -dimensional KP-Boussinesq equation

NASA Astrophysics Data System (ADS)

Sun, Baonan; Wazwaz, Abdul-Majid

2018-11-01

In this work, we investigate the (3 + 1) -dimensional KP-Boussinesq equation, which can be used to describe the nonlinear dynamic behavior in scientific and engineering applications. We derive general high-order soliton solutions by using the Hirota's bilinear method combined with the perturbation expansion technique. We also obtain periodic solutions comprising of high-order breathers, periodic line waves, and mixed solutions consisting of breathers and periodic line waves upon selecting particular parameter constraints of the obtained soliton solutions. Furthermore, smooth rational solutions are generated by taking a long wave limit of the soliton solutions. These smooth rational solutions include high-order rogue waves, high-order lumps, and hybrid solutions consisting of lumps and line rogue waves. To better understand the dynamical behaviors of these solutions, we discuss some illustrative graphical analyses. It is expected that our results can enrich the dynamical behavior of the (3 + 1) -dimensional nonlinear evolution equations of other forms.

Bifurcation analysis for ion acoustic waves in a strongly coupled plasma including trapped electrons

NASA Astrophysics Data System (ADS)

El-Labany, S. K.; El-Taibany, W. F.; Atteya, A.

2018-02-01

The nonlinear ion acoustic wave propagation in a strongly coupled plasma composed of ions and trapped electrons has been investigated. The reductive perturbation method is employed to derive a modified Korteweg-de Vries-Burgers (mKdV-Burgers) equation. To solve this equation in case of dissipative system, the tangent hyperbolic method is used, and a shock wave solution is obtained. Numerical investigations show that, the ion acoustic waves are significantly modified by the effect of polarization force, the trapped electrons and the viscosity coefficients. Applying the bifurcation theory to the dynamical system of the derived mKdV-Burgers equation, the phase portraits of the traveling wave solutions of both of dissipative and non-dissipative systems are analyzed. The present results could be helpful for a better understanding of the waves nonlinear propagation in a strongly coupled plasma, which can be produced by photoionizing laser-cooled and trapped electrons [1], and also in neutron stars or white dwarfs interior.

Dissipative quantum trajectories in complex space: Damped harmonic oscillator

DOE Office of Scientific and Technical Information (OSTI.GOV)

Chou, Chia-Chun, E-mail: ccchou@mx.nthu.edu.tw

Dissipative quantum trajectories in complex space are investigated in the framework of the logarithmic nonlinear Schrödinger equation. The logarithmic nonlinear Schrödinger equation provides a phenomenological description for dissipative quantum systems. Substituting the wave function expressed in terms of the complex action into the complex-extended logarithmic nonlinear Schrödinger equation, we derive the complex quantum Hamilton–Jacobi equation including the dissipative potential. It is shown that dissipative quantum trajectories satisfy a quantum Newtonian equation of motion in complex space with a friction force. Exact dissipative complex quantum trajectories are analyzed for the wave and solitonlike solutions to the logarithmic nonlinear Schrödinger equation formore » the damped harmonic oscillator. These trajectories converge to the equilibrium position as time evolves. It is indicated that dissipative complex quantum trajectories for the wave and solitonlike solutions are identical to dissipative complex classical trajectories for the damped harmonic oscillator. This study develops a theoretical framework for dissipative quantum trajectories in complex space.« less

Particles, Waves, and the Interpretation of Quantum Mechanics

ERIC Educational Resources Information Center

Christoudouleas, N. D.

1975-01-01

Presents an explanation, without mathematical equations, of the basic principles of quantum mechanics. Includes wave-particle duality, the probability character of the wavefunction, and the uncertainty relations. (MLH)

Parabolic equation for nonlinear acoustic wave propagation in inhomogeneous moving media

NASA Astrophysics Data System (ADS)

Aver'yanov, M. V.; Khokhlova, V. A.; Sapozhnikov, O. A.; Blanc-Benon, Ph.; Cleveland, R. O.

2006-12-01

A new parabolic equation is derived to describe the propagation of nonlinear sound waves in inhomogeneous moving media. The equation accounts for diffraction, nonlinearity, absorption, scalar inhomogeneities (density and sound speed), and vectorial inhomogeneities (flow). A numerical algorithm employed earlier to solve the KZK equation is adapted to this more general case. A two-dimensional version of the algorithm is used to investigate the propagation of nonlinear periodic waves in media with random inhomogeneities. For the case of scalar inhomogeneities, including the case of a flow parallel to the wave propagation direction, a complex acoustic field structure with multiple caustics is obtained. Inclusion of the transverse component of vectorial random inhomogeneities has little effect on the acoustic field. However, when a uniform transverse flow is present, the field structure is shifted without changing its morphology. The impact of nonlinearity is twofold: it produces strong shock waves in focal regions, while, outside the caustics, it produces higher harmonics without any shocks. When the intensity is averaged across the beam propagating through a random medium, it evolves similarly to the intensity of a plane nonlinear wave, indicating that the transverse redistribution of acoustic energy gives no considerable contribution to nonlinear absorption.

Experimentally validated multiphysics computational model of focusing and shock wave formation in an electromagnetic lithotripter.

PubMed

Fovargue, Daniel E; Mitran, Sorin; Smith, Nathan B; Sankin, Georgy N; Simmons, Walter N; Zhong, Pei

2013-08-01

A multiphysics computational model of the focusing of an acoustic pulse and subsequent shock wave formation that occurs during extracorporeal shock wave lithotripsy is presented. In the electromagnetic lithotripter modeled in this work the focusing is achieved via a polystyrene acoustic lens. The transition of the acoustic pulse through the solid lens is modeled by the linear elasticity equations and the subsequent shock wave formation in water is modeled by the Euler equations with a Tait equation of state. Both sets of equations are solved simultaneously in subsets of a single computational domain within the BEARCLAW framework which uses a finite-volume Riemann solver approach. This model is first validated against experimental measurements with a standard (or original) lens design. The model is then used to successfully predict the effects of a lens modification in the form of an annular ring cut. A second model which includes a kidney stone simulant in the domain is also presented. Within the stone the linear elasticity equations incorporate a simple damage model.

Experimentally validated multiphysics computational model of focusing and shock wave formation in an electromagnetic lithotripter

PubMed Central

Fovargue, Daniel E.; Mitran, Sorin; Smith, Nathan B.; Sankin, Georgy N.; Simmons, Walter N.; Zhong, Pei

2013-01-01

A multiphysics computational model of the focusing of an acoustic pulse and subsequent shock wave formation that occurs during extracorporeal shock wave lithotripsy is presented. In the electromagnetic lithotripter modeled in this work the focusing is achieved via a polystyrene acoustic lens. The transition of the acoustic pulse through the solid lens is modeled by the linear elasticity equations and the subsequent shock wave formation in water is modeled by the Euler equations with a Tait equation of state. Both sets of equations are solved simultaneously in subsets of a single computational domain within the BEARCLAW framework which uses a finite-volume Riemann solver approach. This model is first validated against experimental measurements with a standard (or original) lens design. The model is then used to successfully predict the effects of a lens modification in the form of an annular ring cut. A second model which includes a kidney stone simulant in the domain is also presented. Within the stone the linear elasticity equations incorporate a simple damage model. PMID:23927200

Waves in magnetized quark matter

NASA Astrophysics Data System (ADS)

Fogaça, D. A.; Sanches, S. M.; Navarra, F. S.

2018-05-01

We study wave propagation in a non-relativistic cold quark-gluon plasma immersed in a constant magnetic field. Starting from the Euler equation we derive linear wave equations and investigate their stability and causality. We use a generic form for the equation of state, the EOS derived from the MIT bag model and also a variant of the this model which includes gluon degrees of freedom. The results of this analysis may be relevant for perturbations propagating through the quark matter phase in the core of compact stars and also for perturbations propagating in the low temperature quark-gluon plasma formed in low energy heavy ion collisions, to be carried out at FAIR and NICA.

The effect of convection and shear on the damping and propagation of pressure waves

NASA Astrophysics Data System (ADS)

Kiel, Barry Vincent

Combustion instability is the positive feedback between heat release and pressure in a combustion system. Combustion instability occurs in the both air breathing and rocket propulsion devices, frequently resulting in high amplitude spinning waves. If unchecked, the resultant pressure fluctuations can cause significant damage. Models for the prediction of combustion instability typically include models for the heat release, the wave propagation and damping. Many wave propagation models for propulsion systems assume negligible flow, resulting in the wave equation. In this research the effect of flow on wave propagation was studied both numerically and experimentally. Two experiential rigs were constructed, one with axial flow to study the longitudinal waves, the other with swirling flow to study circumferential waves. The rigs were excited with speakers and the resultant pressure was measured simultaneously at many locations. Models of the rig were also developed. Equations for wave propagation were derived from the Euler Equations. The resultant resembled the wave equation with three additional terms, two for the effect of the convection and a one for the effect of shear of the mean flow on wave propagation. From the experimental and numerical data several conclusions were made. First, convection and shear both act as damping on the wave propagation, reducing the magnitude of the Frequency Response Function and the resonant frequency of the modes. Second, the energy extracted from the mean flow as a result of turbulent shear for a given condition is frequency dependent, decreasing with increasing frequency. The damping of the modes, measured for the same shear flow, also decreased with frequency. Finally, the two convective terms cause the anti-nodes of the modes to no longer be stationary. For both the longitudinal and circumferential waves, the anti-nodes move through the domain even for mean flow Mach numbers less than 0.10. It was concluded that convection causes the spinning waves documented in inlets and exhausts of gas turbine engines, rocket combustion chambers, and afterburner chambers. As a result, the effects of shear must be included when modeling wave propagation, even for mean flows less than < Mach 0.10.

Hydrodynamic and kinetic models for spin-1/2 electron-positron quantum plasmas: Annihilation interaction, helicity conservation, and wave dispersion in magnetized plasmas

DOE Office of Scientific and Technical Information (OSTI.GOV)

Andreev, Pavel A., E-mail: andreevpa@physics.msu.ru

2015-06-15

We discuss the complete theory of spin-1/2 electron-positron quantum plasmas, when electrons and positrons move with velocities mach smaller than the speed of light. We derive a set of two fluid quantum hydrodynamic equations consisting of the continuity, Euler, spin (magnetic moment) evolution equations for each species. We explicitly include the Coulomb, spin-spin, Darwin and annihilation interactions. The annihilation interaction is the main topic of the paper. We consider the contribution of the annihilation interaction in the quantum hydrodynamic equations and in the spectrum of waves in magnetized electron-positron plasmas. We consider the propagation of waves parallel and perpendicular tomore » an external magnetic field. We also consider the oblique propagation of longitudinal waves. We derive the set of quantum kinetic equations for electron-positron plasmas with the Darwin and annihilation interactions. We apply the kinetic theory to the linear wave behavior in absence of external fields. We calculate the contribution of the Darwin and annihilation interactions in the Landau damping of the Langmuir waves. We should mention that the annihilation interaction does not change number of particles in the system. It does not related to annihilation itself, but it exists as a result of interaction of an electron-positron pair via conversion of the pair into virtual photon. A pair of the non-linear Schrodinger equations for the electron-positron plasmas including the Darwin and annihilation interactions is derived. Existence of the conserving helicity in electron-positron quantum plasmas of spinning particles with the Darwin and annihilation interactions is demonstrated. We show that the annihilation interaction plays an important role in the quantum electron-positron plasmas giving the contribution of the same magnitude as the spin-spin interaction.« less

Mathematical nonlinear optics

NASA Astrophysics Data System (ADS)

McLaughlin, David W.

1995-08-01

The principal investigator, together with a post-doctoral fellows Tetsuji Ueda and Xiao Wang, several graduate students, and colleagues, has applied the modern mathematical theory of nonlinear waves to problems in nonlinear optics and to equations directly relevant to nonlinear optics. Projects included the interaction of laser light with nematic liquid crystals and chaotic, homoclinic, small dispersive, and random behavior of solutions of the nonlinear Schroedinger equation. In project 1, the extremely strong nonlinear response of a continuous wave laser beam in a nematic liquid crystal medium has produced striking undulation and filamentation of the laser beam which has been observed experimentally and explained theoretically. In project 2, qualitative properties of the nonlinear Schroedinger equation (which is the fundamental equation for nonlinear optics) have been identified and studied. These properties include optical shocking behavior in the limit of very small dispersion, chaotic and homoclinic behavior in discretizations of the partial differential equation, and random behavior.

Acceleration of stable TTI P-wave reverse-time migration with GPUs

NASA Astrophysics Data System (ADS)

Kim, Youngseo; Cho, Yongchae; Jang, Ugeun; Shin, Changsoo

2013-03-01

When a pseudo-acoustic TTI (tilted transversely isotropic) coupled wave equation is used to implement reverse-time migration (RTM), shear wave energy is significantly included in the migration image. Because anisotropy has intrinsic elastic characteristics, coupling P-wave and S-wave modes in the pseudo-acoustic wave equation is inevitable. In RTM with only primary energy or the P-wave mode in seismic data, the S-wave energy is regarded as noise for the migration image. To solve this problem, we derive a pure P-wave equation for TTI media that excludes the S-wave energy. Additionally, we apply the rapid expansion method (REM) based on a Chebyshev expansion and a pseudo-spectral method (PSM) to calculate spatial derivatives in the wave equation. When REM is incorporated with the PSM for the spatial derivatives, wavefields with high numerical accuracy can be obtained without grid dispersion when performing numerical wave modeling. Another problem in the implementation of TTI RTM is that wavefields in an area with high gradients of dip or azimuth angles can be blown up in the progression of the forward and backward algorithms of the RTM. We stabilize the wavefields by applying a spatial-frequency domain high-cut filter when calculating the spatial derivatives using the PSM. In addition, to increase performance speed, the graphic processing unit (GPU) architecture is used instead of traditional CPU architecture. To confirm the degree of acceleration compared to the CPU version on our RTM, we then analyze the performance measurements according to the number of GPUs employed.

Expansion shock waves in regularized shallow-water theory

NASA Astrophysics Data System (ADS)

El, Gennady A.; Hoefer, Mark A.; Shearer, Michael

2016-05-01

We identify a new type of shock wave by constructing a stationary expansion shock solution of a class of regularized shallow-water equations that include the Benjamin-Bona-Mahony and Boussinesq equations. An expansion shock exhibits divergent characteristics, thereby contravening the classical Lax entropy condition. The persistence of the expansion shock in initial value problems is analysed and justified using matched asymptotic expansions and numerical simulations. The expansion shock's existence is traced to the presence of a non-local dispersive term in the governing equation. We establish the algebraic decay of the shock as it is gradually eroded by a simple wave on either side. More generally, we observe a robustness of the expansion shock in the presence of weak dissipation and in simulations of asymmetric initial conditions where a train of solitary waves is shed from one side of the shock.

A shallow water model for the propagation of tsunami via Lattice Boltzmann method

NASA Astrophysics Data System (ADS)

Zergani, Sara; Aziz, Z. A.; Viswanathan, K. K.

2015-01-01

An efficient implementation of the lattice Boltzmann method (LBM) for the numerical simulation of the propagation of long ocean waves (e.g. tsunami), based on the nonlinear shallow water (NSW) wave equation is presented. The LBM is an alternative numerical procedure for the description of incompressible hydrodynamics and has the potential to serve as an efficient solver for incompressible flows in complex geometries. This work proposes the NSW equations for the irrotational surface waves in the case of complex bottom elevation. In recent time, equation involving shallow water is the current norm in modelling tsunami operations which include the propagation zone estimation. Several test-cases are presented to verify our model. Some implications to tsunami wave modelling are also discussed. Numerical results are found to be in excellent agreement with theory.

Inverse random source scattering for the Helmholtz equation in inhomogeneous media

NASA Astrophysics Data System (ADS)

Li, Ming; Chen, Chuchu; Li, Peijun

2018-01-01

This paper is concerned with an inverse random source scattering problem in an inhomogeneous background medium. The wave propagation is modeled by the stochastic Helmholtz equation with the source driven by additive white noise. The goal is to reconstruct the statistical properties of the random source such as the mean and variance from the boundary measurement of the radiated random wave field at multiple frequencies. Both the direct and inverse problems are considered. We show that the direct problem has a unique mild solution by a constructive proof. For the inverse problem, we derive Fredholm integral equations, which connect the boundary measurement of the radiated wave field with the unknown source function. A regularized block Kaczmarz method is developed to solve the ill-posed integral equations. Numerical experiments are included to demonstrate the effectiveness of the proposed method.

Unimodal dynamical systems: Comparison principles, spreading speeds and travelling waves

NASA Astrophysics Data System (ADS)

Yi, Taishan; Chen, Yuming; Wu, Jianhong

Reaction diffusion equations with delayed nonlinear reaction terms are used as prototypes to motivate an appropriate abstract formulation of dynamical systems with unimodal nonlinearity. For such non-monotone dynamical systems, we develop a general comparison principle and show how this general comparison principle, coupled with some existing results for monotone dynamical systems, can be used to establish results on the asymptotic speeds of spread and travelling waves. We illustrate our main results by an integral equation which includes a nonlocal delayed reaction diffusion equation and a nonlocal delayed lattice differential system in an unbounded domain, with the non-monotone nonlinearities including the Ricker birth function and the Mackey-Glass hematopoiesis feedback.

Asymptotic problems for stochastic partial differential equations

NASA Astrophysics Data System (ADS)

Salins, Michael

Stochastic partial differential equations (SPDEs) can be used to model systems in a wide variety of fields including physics, chemistry, and engineering. The main SPDEs of interest in this dissertation are the semilinear stochastic wave equations which model the movement of a material with constant mass density that is exposed to both determinstic and random forcing. Cerrai and Freidlin have shown that on fixed time intervals, as the mass density of the material approaches zero, the solutions of the stochastic wave equation converge uniformly to the solutions of a stochastic heat equation, in probability. This is called the Smoluchowski-Kramers approximation. In Chapter 2, we investigate some of the multi-scale behaviors that these wave equations exhibit. In particular, we show that the Freidlin-Wentzell exit place and exit time asymptotics for the stochastic wave equation in the small noise regime can be approximated by the exit place and exit time asymptotics for the stochastic heat equation. We prove that the exit time and exit place asymptotics are characterized by quantities called quasipotentials and we prove that the quasipotentials converge. We then investigate the special case where the equation has a gradient structure and show that we can explicitly solve for the quasipotentials, and that the quasipotentials for the heat equation and wave equation are equal. In Chapter 3, we study the Smoluchowski-Kramers approximation in the case where the material is electrically charged and exposed to a magnetic field. Interestingly, if the system is frictionless, then the Smoluchowski-Kramers approximation does not hold. We prove that the Smoluchowski-Kramers approximation is valid for systems exposed to both a magnetic field and friction. Notably, we prove that the solutions to the second-order equations converge to the solutions of the first-order equation in an Lp sense. This strengthens previous results where convergence was proved in probability.

Multiple periodic-soliton solutions of the (3+1)-dimensional generalised shallow water equation

NASA Astrophysics Data System (ADS)

Li, Ye-Zhou; Liu, Jian-Guo

2018-06-01

Based on the extended variable-coefficient homogeneous balance method and two new ansätz functions, we construct auto-Bäcklund transformation and multiple periodic-soliton solutions of (3 {+} 1)-dimensional generalised shallow water equations. Completely new periodic-soliton solutions including periodic cross-kink wave, periodic two-solitary wave and breather type of two-solitary wave are obtained. In addition, cross-kink three-soliton and cross-kink four-soliton solutions are derived. Furthermore, propagation characteristics and interactions of the obtained solutions are discussed and illustrated in figures.

Modeling of Nonlinear Hydrodynamics of the Coastal Areas of the Black Sea by the Chain of the Proprietary and Open Source Models

NASA Astrophysics Data System (ADS)

Kantardgi, Igor; Zheleznyak, Mark; Demchenko, Raisa; Dykyi, Pavlo; Kivva, Sergei; Kolomiets, Pavlo; Sorokin, Maxim

2014-05-01

The nearshore hydrodynamic fields are produced by the nonlinear interactions of the shoaling waves of different time scales and currents. To simulate the wind wave and swells propagated to the coasts, wave generated near shore currents, nonlinear-dispersive wave transformation and wave diffraction in interaction with coastal and port structure, sediment transport and coastal erosion the chains of the models should be used. The objective of this presentation is to provide an overview of the results of the application of the model chains for the assessment of the wave impacts on new construction designed at the Black Sea coasts and the impacts of these constructions on the coastal erosion/ accretion processes to demonstrate needs for further development of the nonlinear models for the coastal engineering applications. The open source models Wave Watch III and SWAN has been used to simulate wave statistics of the dedicated areas of the Black Sea in high resolution to calculated the statistical parameters of the extreme wave approaching coastal zone construction in accordance with coastal engineering standards. As the main tool for the costal hydrodynamic simulations the modeling system COASTOX-MORPHO has been used, that includes the following models. HWAVE -code based on hyperbolic version of mild slope equations., HWAVE-S - spectral version of HWAVE., BOUSS-FNL - fully nonlinear system of Boussinesq equations for simulation wave nonlinear -dispersive wave transformation in coastal areas. COASTOX-CUR - the code provided the numerical solution of the Nonlinear Shallow Water Equations (NLSWE) by finite-volume methods on the unstructured grid describing the long wave transformation in the coastal zone with the efficient drying -wetting algorithms to simulate the inundation of the coastal areas including tsunami wave runup. Coastox -Cur equations with the radiation stress term calculated via near shore wave fields simulate the wave generated nearhore currents. COASTOX-SED - the module of the simulation of the sediment transport in which the suspended sediments are simulated on the basis of the solution of 2-D advection -diffusion equation and the bottom sediment transport calculations are provided the basis of a library of the most popular semi-empirical formulas. MORPH - the module of the simulation of the morphological transformation of coastal zone based on the mass balance equation, on the basis of the sediment fluxes, calculated in the SED module. MORPH management submodel is responsible for the execution of the model chain "waves- current- sediments - morphodynamics- waves". The open source model SWASH has been used to simulate nonlinear resonance phenomena in coastal waters. The model chain was applied to simulate the potential impact of the designed shore protection structures at the Sochi Olympic Park on coastal morphodynamics, the wave parameters and nonlinear oscillations in the new ports designed in Gelenddjik and Taman at North-East coast of the Black Sea. The modeling results are compared with the results of the physical modeling in the hydraulic flumes of Moscow University of Civil Engineering.

Local algebraic analysis of differential systems

NASA Astrophysics Data System (ADS)

Kaptsov, O. V.

2015-06-01

We propose a new approach for studying the compatibility of partial differential equations. This approach is a synthesis of the Riquier method, Gröbner basis theory, and elements of algebraic geometry. As applications, we consider systems including the wave equation and the sine-Gordon equation.

New soliton solution to the longitudinal wave equation in a magneto-electro-elastic circular rod

NASA Astrophysics Data System (ADS)

Seadawy, Aly R.; Manafian, Jalil

2018-03-01

This paper examines the effectiveness of an integration scheme which called the extended trial equation method (ETEM) in exactly solving a well-known nonlinear equation of partial differential equations (PDEs). In this respect, the longitudinal wave equation (LWE) that arises in mathematical physics with dispersion caused by the transverse Poisson's effect in a magneto-electro-elastic (MEE) circular rod, which a series of exact traveling wave solutions for the aforementioned equation is formally extracted. Explicit new exact solutions are derived in different form such as dark solitons, bright solitons, solitary wave, periodic solitary wave, rational function, and elliptic function solutions of the longitudinal wave equation. The movements of obtained solutions are shown graphically, which helps to understand the physical phenomena of this longitudinal wave equation. Many other such types of nonlinear equations arising in non-destructive evaluation of structures made of the advanced MEE material can also be solved by this method.

Solitary waves, rogue waves and homoclinic breather waves for a (2 + 1)-dimensional generalized Kadomtsev-Petviashvili equation

NASA Astrophysics Data System (ADS)

Dong, Min-Jie; Tian, Shou-Fu; Yan, Xue-Wei; Zou, Li; Li, Jin

2017-10-01

We study a (2 + 1)-dimensional generalized Kadomtsev-Petviashvili (gKP) equation, which characterizes the formation of patterns in liquid drops. By using Bell’s polynomials, an effective way is employed to succinctly construct the bilinear form of the gKP equation. Based on the resulting bilinear equation, we derive its solitary waves, rogue waves and homoclinic breather waves, respectively. Our results can help enrich the dynamical behavior of the KP-type equations.

Wave theory in rotating systems: Schrödinger equations bridge the gaps between the equatorial β-plane and the spherical earth

NASA Astrophysics Data System (ADS)

Paldor, N.

2017-12-01

The concise and elegant wave theory developed on the equatorial β-plane by Matsuno (1966, M66 hereafter) is based on the formulation of a Schrödinger equation associated with the governing Linear Rotating Shallow Water Equations (LRSWE). The theory yields explicit expressions for the dispersion relations and meridional amplitude structures of all zonally propagating waves - Rossby, Inertia-Gravity, Kelvin and Yanai. In contrast, the spherical wave theory of Longuet-Higgins (1968) is a collection of asymptotic expansions in many sub-ranges e.g. large, small (and even negative) Lamb Number; high and low frequency; low-latitudes, etc. that rests upon extensive numerical solutions of several Ordinary Differential Equations. The difference between the two theories is highlighted by their lengths. The essential elements of the former planar study are completely revealed in just 3-4 pages including the derivation of explicit formulae for the phase speeds and amplitude meridional structures. In comtrast, the latter spherical theory contains 97 pages and the results of the numerical calculations are summarized in 30 pages of tables filled with numerical values and about 31 figures, each of which containing many separate curves! In my talk I will re-visit the wave problem on a sphere by developing several Schrödinger equations that approximate the governing eigenvalue equation associated with zonally propagating waves. Each of the Schrödinger equations approximates the original second order Ordinary Differential Equation in a different range of the 3 parameters: Lamb-Number, frequency and zonal wavenumber. As in M66, each of the Schrödinger equations yields explicit expressions for the dispersion relations and meridional amplitude structure of Rossby and Inertia-Gravity waves. In addition, the analysis shows that Yanai wave exists on a sphere even tough the zonal velocity is regular everywhere there (in contrast to the β-plane where the zonal velocity is singular everywhere) and that Kelvin waves do not exist as a separate mode (but the eastward propagating n=0 Inertia-Gravity is nearly non-dispersive). References Longuet-Higgins, M. S. Phil. Trans. Roy. Soc. London; 262, 511-607; 1968 Matsuno, T.; J. Met. Soc. Japan. 44(1), 25-43; 1966

Rogue waves in the two dimensional nonlocal nonlinear Schrödinger equation and nonlocal Klein-Gordon equation.

PubMed

Liu, Wei; Zhang, Jing; Li, Xiliang

2018-01-01

In this paper, we investigate two types of nonlocal soliton equations with the parity-time (PT) symmetry, namely, a two dimensional nonlocal nonlinear Schrödinger (NLS) equation and a coupled nonlocal Klein-Gordon equation. Solitons and periodic line waves as exact solutions of these two nonlocal equations are derived by employing the Hirota's bilinear method. Like the nonlocal NLS equation, these solutions may have singularities. However, by suitable constraints of parameters, nonsingular breather solutions are generated. Besides, by taking a long wave limit of these obtained soliton solutions, rogue wave solutions and semi-rational solutions are derived. For the two dimensional NLS equation, rogue wave solutions are line rogue waves, which arise from a constant background with a line profile and then disappear into the same background. The semi-rational solutions shows intriguing dynamical behaviours: line rogue wave and line breather arise from a constant background together and then disappear into the constant background again uniformly. For the coupled nonlocal Klein-Gordon equation, rogue waves are localized in both space and time, semi-rational solutions are composed of rogue waves, breathers and periodic line waves. These solutions are demonstrated analytically to exist for special classes of nonlocal equations relevant to optical waveguides.

Rogue waves in the two dimensional nonlocal nonlinear Schrödinger equation and nonlocal Klein-Gordon equation

PubMed Central

Zhang, Jing; Li, Xiliang

2018-01-01

In this paper, we investigate two types of nonlocal soliton equations with the parity-time (PT) symmetry, namely, a two dimensional nonlocal nonlinear Schrödinger (NLS) equation and a coupled nonlocal Klein-Gordon equation. Solitons and periodic line waves as exact solutions of these two nonlocal equations are derived by employing the Hirota’s bilinear method. Like the nonlocal NLS equation, these solutions may have singularities. However, by suitable constraints of parameters, nonsingular breather solutions are generated. Besides, by taking a long wave limit of these obtained soliton solutions, rogue wave solutions and semi-rational solutions are derived. For the two dimensional NLS equation, rogue wave solutions are line rogue waves, which arise from a constant background with a line profile and then disappear into the same background. The semi-rational solutions shows intriguing dynamical behaviours: line rogue wave and line breather arise from a constant background together and then disappear into the constant background again uniformly. For the coupled nonlocal Klein-Gordon equation, rogue waves are localized in both space and time, semi-rational solutions are composed of rogue waves, breathers and periodic line waves. These solutions are demonstrated analytically to exist for special classes of nonlocal equations relevant to optical waveguides. PMID:29432495

Theory and observation of electromagnetic ion cyclotron triggered emissions in the magnetosphere

NASA Astrophysics Data System (ADS)

Omura, Yoshiharu; Pickett, Jolene; Grison, Benjamin; Santolik, Ondrej; Dandouras, Iannis; Engebretson, Mark; Décréau, Pierrette M. E.; Masson, Arnaud

2010-07-01

We develop a nonlinear wave growth theory of electromagnetic ion cyclotron (EMIC) triggered emissions observed in the inner magnetosphere. We first derive the basic wave equations from Maxwell's equations and the momentum equations for the electrons and ions. We then obtain equations that describe the nonlinear dynamics of resonant protons interacting with an EMIC wave. The frequency sweep rate of the wave plays an important role in forming the resonant current that controls the wave growth. Assuming an optimum condition for the maximum growth rate as an absolute instability at the magnetic equator and a self-sustaining growth condition for the wave propagating from the magnetic equator, we obtain a set of ordinary differential equations that describe the nonlinear evolution of a rising tone emission generated at the magnetic equator. Using the physical parameters inferred from the wave, particle, and magnetic field data measured by the Cluster spacecraft, we determine the dispersion relation for the EMIC waves. Integrating the differential equations numerically, we obtain a solution for the time variation of the amplitude and frequency of a rising tone emission at the equator. Assuming saturation of the wave amplitude, as is found in the observations, we find good agreement between the numerical solutions and the wave spectrum of the EMIC triggered emissions.

Interactions of localized wave structures and dynamics in the defocusing coupled nonlinear Schrödinger equations.

PubMed

Zhang, Guoqiang; Yan, Zhenya; Wen, Xiao-Yong; Chen, Yong

2017-04-01

We investigate the defocusing coupled nonlinear Schrödinger equations from a 3×3 Lax pair. The Darboux transformations with the nonzero plane-wave solutions are presented to derive the newly localized wave solutions including dark-dark and bright-dark solitons, breather-breather solutions, and different types of new vector rogue wave solutions, as well as interactions between distinct types of localized wave solutions. Moreover, we analyze these solutions by means of parameters modulation. Finally, the perturbed wave propagations of some obtained solutions are explored by means of systematic simulations, which demonstrates that nearly stable and strongly unstable solutions. Our research results could constitute a significant contribution to explore the distinct nonlinear waves (e.g., dark solitons, breather solutions, and rogue wave solutions) dynamics of the coupled system in related fields such as nonlinear optics, plasma physics, oceanography, and Bose-Einstein condensates.

A numerical study of the 3-periodic wave solutions to KdV-type equations

NASA Astrophysics Data System (ADS)

Zhang, Yingnan; Hu, Xingbiao; Sun, Jianqing

2018-02-01

In this paper, by using the direct method of calculating periodic wave solutions proposed by Akira Nakamura, we present a numerical process to calculate the 3-periodic wave solutions to several KdV-type equations: the Korteweg-de Vries equation, the Sawada-Koterra equation, the Boussinesq equation, the Ito equation, the Hietarinta equation and the (2 + 1)-dimensional Kadomtsev-Petviashvili equation. Some detailed numerical examples are given to show the existence of the three-periodic wave solutions numerically.

Spatio-temporal dynamics of an active, polar, viscoelastic ring.

PubMed

Marcq, Philippe

2014-04-01

Constitutive equations for a one-dimensional, active, polar, viscoelastic liquid are derived by treating the strain field as a slow hydrodynamic variable. Taking into account the couplings between strain and polarity allowed by symmetry, the hydrodynamics of an active, polar, viscoelastic body include an evolution equation for the polarity field that generalizes the damped Kuramoto-Sivashinsky equation. Beyond thresholds of the active coupling coefficients between the polarity and the stress or the strain rate, bifurcations of the homogeneous state lead first to stationary waves, then to propagating waves of the strain, stress and polarity fields. I argue that these results are relevant to living matter, and may explain rotating actomyosin rings in cells and mechanical waves in epithelial cell monolayers.

Effect of surface wave propagation in a four-layered oceanic crust model

NASA Astrophysics Data System (ADS)

Paul, Pasupati; Kundu, Santimoy; Mandal, Dinbandhu

2017-12-01

Dispersion of Rayleigh type surface wave propagation has been discussed in four-layered oceanic crust. It includes a sandy layer over a crystalline elastic half-space and over it there are two more layers—on the top inhomogeneous liquid layer and under it a liquid-saturated porous layer. Frequency equation is obtained in the form of determinant. The effects of the width of different layers as well as the inhomogeneity of liquid layer, sandiness of sandy layer on surface waves are depicted and shown graphically by considering all possible case of the particular model. Some special cases have been deduced, few special cases give the dispersion equation of Scholte wave and Stoneley wave, some of which have already been discussed elsewhere.

Unsplit complex frequency shifted perfectly matched layer for second-order wave equation using auxiliary differential equations.

PubMed

Gao, Yingjie; Zhang, Jinhai; Yao, Zhenxing

2015-12-01

The complex frequency shifted perfectly matched layer (CFS-PML) can improve the absorbing performance of PML for nearly grazing incident waves. However, traditional PML and CFS-PML are based on first-order wave equations; thus, they are not suitable for second-order wave equation. In this paper, an implementation of CFS-PML for second-order wave equation is presented using auxiliary differential equations. This method is free of both convolution calculations and third-order temporal derivatives. As an unsplit CFS-PML, it can reduce the nearly grazing incidence. Numerical experiments show that it has better absorption than typical PML implementations based on second-order wave equation.

Higher-order rational solitons and rogue-like wave solutions of the (2 + 1)-dimensional nonlinear fluid mechanics equations

NASA Astrophysics Data System (ADS)

Wen, Xiao-Yong; Yan, Zhenya

2017-02-01

The novel generalized perturbation (n, M)-fold Darboux transformations (DTs) are reported for the (2 + 1)-dimensional Kadomtsev-Petviashvili (KP) equation and its extension by using the Taylor expansion of the Darboux matrix. The generalized perturbation (1 , N - 1) -fold DTs are used to find their higher-order rational solitons and rogue wave solutions in terms of determinants. The dynamics behaviors of these rogue waves are discussed in detail for different parameters and time, which display the interesting RW and soliton structures including the triangle, pentagon, heptagon profiles, etc. Moreover, we find that a new phenomenon that the parameter (a) can control the wave structures of the KP equation from the higher-order rogue waves (a ≠ 0) into higher-order rational solitons (a = 0) in (x, t)-space with y = const . These results may predict the corresponding dynamical phenomena in the models of fluid mechanics and other physically relevant systems.

Modified Korteweg–de Vries equation in a negative ion rich hot adiabatic dusty plasma with non-thermal ion and trapped electron

DOE Office of Scientific and Technical Information (OSTI.GOV)

Adhikary, N. C., E-mail: nirab-iasst@yahoo.co.in; Deka, M. K.; Dev, A. N.

2014-08-15

In this report, the investigation of the properties of dust acoustic (DA) solitary wave propagation in an adiabatic dusty plasma including the effect of the non-thermal ions and trapped electrons is presented. The reductive perturbation method has been employed to derive the modified Korteweg–de Vries (mK-dV) equation for dust acoustic solitary waves in a homogeneous, unmagnetized, and collisionless plasma whose constituents are electrons, singly charged positive ions, singly charged negative ions, and massive charged dust particles. The stationary analytical solution of the mK-dV equation is numerically analyzed and where the effect of various dusty plasma constituents DA solitary wave propagationmore » is taken into account. It is observed that both the ions in dusty plasma play as a key role for the formation of both rarefactive as well as the compressive DA solitary waves and also the ion concentration controls the transformation of negative to positive potentials of the waves.« less

Nonlinear acoustic wave equations with fractional loss operators.

PubMed

Prieur, Fabrice; Holm, Sverre

2011-09-01

Fractional derivatives are well suited to describe wave propagation in complex media. When introduced in classical wave equations, they allow a modeling of attenuation and dispersion that better describes sound propagation in biological tissues. Traditional constitutive equations from solid mechanics and heat conduction are modified using fractional derivatives. They are used to derive a nonlinear wave equation which describes attenuation and dispersion laws that match observations. This wave equation is a generalization of the Westervelt equation, and also leads to a fractional version of the Khokhlov-Zabolotskaya-Kuznetsov and Burgers' equations. © 2011 Acoustical Society of America

Classifying bilinear differential equations by linear superposition principle

NASA Astrophysics Data System (ADS)

Zhang, Lijun; Khalique, Chaudry Masood; Ma, Wen-Xiu

2016-09-01

In this paper, we investigate the linear superposition principle of exponential traveling waves to construct a sub-class of N-wave solutions of Hirota bilinear equations. A necessary and sufficient condition for Hirota bilinear equations possessing this specific sub-class of N-wave solutions is presented. We apply this result to find N-wave solutions to the (2+1)-dimensional KP equation, a (3+1)-dimensional generalized Kadomtsev-Petviashvili (KP) equation, a (3+1)-dimensional generalized BKP equation and the (2+1)-dimensional BKP equation. The inverse question, i.e., constructing Hirota Bilinear equations possessing N-wave solutions, is considered and a refined 3-step algorithm is proposed. As examples, we construct two very general kinds of Hirota bilinear equations of order 4 possessing N-wave solutions among which one satisfies dispersion relation and another does not satisfy dispersion relation.

Undular bore theory for the Gardner equation

NASA Astrophysics Data System (ADS)

Kamchatnov, A. M.; Kuo, Y.-H.; Lin, T.-C.; Horng, T.-L.; Gou, S.-C.; Clift, R.; El, G. A.; Grimshaw, R. H. J.

2012-09-01

We develop modulation theory for undular bores (dispersive shock waves) in the framework of the Gardner, or extended Korteweg-de Vries (KdV), equation, which is a generic mathematical model for weakly nonlinear and weakly dispersive wave propagation, when effects of higher order nonlinearity become important. Using a reduced version of the finite-gap integration method we derive the Gardner-Whitham modulation system in a Riemann invariant form and show that it can be mapped onto the well-known modulation system for the Korteweg-de Vries equation. The transformation between the two counterpart modulation systems is, however, not invertible. As a result, the study of the resolution of an initial discontinuity for the Gardner equation reveals a rich phenomenology of solutions which, along with the KdV-type simple undular bores, include nonlinear trigonometric bores, solibores, rarefaction waves, and composite solutions representing various combinations of the above structures. We construct full parametric maps of such solutions for both signs of the cubic nonlinear term in the Gardner equation. Our classification is supported by numerical simulations.

Calculation of the flow field including boundary layer effects for supersonic mixed compression inlets at angles of attack

NASA Technical Reports Server (NTRS)

Vadyak, J.; Hoffman, J. D.

1982-01-01

The flow field in supersonic mixed compression aircraft inlets at angle of attack is calculated. A zonal modeling technique is employed to obtain the solution which divides the flow field into different computational regions. The computational regions consist of a supersonic core flow, boundary layer flows adjacent to both the forebody/centerbody and cowl contours, and flow in the shock wave boundary layer interaction regions. The zonal modeling analysis is described and some computational results are presented. The governing equations for the supersonic core flow form a hyperbolic system of partial differential equations. The equations for the characteristic surfaces and the compatibility equations applicable along these surfaces are derived. The characteristic surfaces are the stream surfaces, which are surfaces composed of streamlines, and the wave surfaces, which are surfaces tangent to a Mach conoid. The compatibility equations are expressed as directional derivatives along streamlines and bicharacteristics, which are the lines of tangency between a wave surface and a Mach conoid.

Traveling waves and conservation laws for highly nonlinear wave equations modeling Hertz chains

NASA Astrophysics Data System (ADS)

Przedborski, Michelle; Anco, Stephen C.

2017-09-01

A highly nonlinear, fourth-order wave equation that models the continuum theory of long wavelength pulses in weakly compressed, homogeneous, discrete chains with a general power-law contact interaction is studied. For this wave equation, all solitary wave solutions and all nonlinear periodic wave solutions, along with all conservation laws, are derived. The solutions are explicitly parameterized in terms of the asymptotic value of the wave amplitude in the case of solitary waves and the peak of the wave amplitude in the case of nonlinear periodic waves. All cases in which the solution expressions can be stated in an explicit analytic form using elementary functions are worked out. In these cases, explicit expressions for the total energy and total momentum for all solutions are obtained as well. The derivation of the solutions uses the conservation laws combined with an energy analysis argument to reduce the wave equation directly to a separable first-order differential equation that determines the wave amplitude in terms of the traveling wave variable. This method can be applied more generally to other highly nonlinear wave equations.

Expansion shock waves in regularized shallow-water theory

PubMed Central

El, Gennady A.; Shearer, Michael

2016-01-01

We identify a new type of shock wave by constructing a stationary expansion shock solution of a class of regularized shallow-water equations that include the Benjamin–Bona–Mahony and Boussinesq equations. An expansion shock exhibits divergent characteristics, thereby contravening the classical Lax entropy condition. The persistence of the expansion shock in initial value problems is analysed and justified using matched asymptotic expansions and numerical simulations. The expansion shock's existence is traced to the presence of a non-local dispersive term in the governing equation. We establish the algebraic decay of the shock as it is gradually eroded by a simple wave on either side. More generally, we observe a robustness of the expansion shock in the presence of weak dissipation and in simulations of asymmetric initial conditions where a train of solitary waves is shed from one side of the shock. PMID:27279780

Transcritical flow of a stratified fluid over topography: analysis of the forced Gardner equation

NASA Astrophysics Data System (ADS)

Kamchatnov, A. M.; Kuo, Y.-H.; Lin, T.-C.; Horng, T.-L.; Gou, S.-C.; Clift, R.; El, G. A.; Grimshaw, R. H. J.

2013-12-01

Transcritical flow of a stratified fluid past a broad localised topographic obstacle is studied analytically in the framework of the forced extended Korteweg--de Vries (eKdV), or Gardner, equation. We consider both possible signs for the cubic nonlinear term in the Gardner equation corresponding to different fluid density stratification profiles. We identify the range of the input parameters: the oncoming flow speed (the Froude number) and the topographic amplitude, for which the obstacle supports a stationary localised hydraulic transition from the subcritical flow upstream to the supercritical flow downstream. Such a localised transcritical flow is resolved back into the equilibrium flow state away from the obstacle with the aid of unsteady coherent nonlinear wave structures propagating upstream and downstream. Along with the regular, cnoidal undular bores occurring in the analogous problem for the single-layer flow modeled by the forced KdV equation, the transcritical internal wave flows support a diverse family of upstream and downstream wave structures, including solibores, rarefaction waves, reversed and trigonometric undular bores, which we describe using the recent development of the nonlinear modulation theory for the (unforced) Gardner equation. The predictions of the developed analytic construction are confirmed by direct numerical simulations of the forced Gardner equation for a broad range of input parameters.

Preserving the Helmholtz dispersion relation: One-way acoustic wave propagation using matrix square roots

NASA Astrophysics Data System (ADS)

Keefe, Laurence

2016-11-01

Parabolized acoustic propagation in transversely inhomogeneous media is described by the operator update equation U (x , y , z + Δz) =eik0 (- 1 +√{ 1 + Z }) U (x , y , z) for evolution of the envelope of a wavetrain solution to the original Helmholtz equation. Here the operator, Z =∇T2 + (n2 - 1) , involves the transverse Laplacian and the refractive index distribution. Standard expansion techniques (on the assumption Z << 1)) produce pdes that approximate, to greater or lesser extent, the full dispersion relation of the original Helmholtz equation, except that none of them describe evanescent/damped waves without special modifications to the expansion coefficients. Alternatively, a discretization of both the envelope and the operator converts the operator update equation into a matrix multiply, and existing theorems on matrix functions demonstrate that the complete (discrete) Helmholtz dispersion relation, including evanescent/damped waves, is preserved by this discretization. Propagation-constant/damping-rates contour comparisons for the operator equation and various approximations demonstrate this point, and how poorly the lowest-order, textbook, parabolized equation describes propagation in lined ducts.

Scattered surface wave energy in the seismic coda

USGS Publications Warehouse

Zeng, Y.

2006-01-01

One of the many important contributions that Aki has made to seismology pertains to the origin of coda waves (Aki, 1969; Aki and Chouet, 1975). In this paper, I revisit Aki's original idea of the role of scattered surface waves in the seismic coda. Based on the radiative transfer theory, I developed a new set of scattered wave energy equations by including scattered surface waves and body wave to surface wave scattering conversions. The work is an extended study of Zeng et al. (1991), Zeng (1993) and Sato (1994a) on multiple isotropic-scattering, and may shed new insight into the seismic coda wave interpretation. The scattering equations are solved numerically by first discretizing the model at regular grids and then solving the linear integral equations iteratively. The results show that scattered wave energy can be well approximated by body-wave to body wave scattering at earlier arrival times and short distances. At long distances from the source, scattered surface waves dominate scattered body waves at surface stations. Since surface waves are 2-D propagating waves, their scattered energies should in theory follow a common decay curve. The observed common decay trends on seismic coda of local earthquake recordings particular at long lapse times suggest that perhaps later seismic codas are dominated by scattered surface waves. When efficient body wave to surface wave conversion mechanisms are present in the shallow crustal layers, such as soft sediment layers, the scattered surface waves dominate the seismic coda at even early arrival times for shallow sources and at later arrival times for deeper events.

Optical rogue waves associated with the negative coherent coupling in an isotropic medium.

PubMed

Sun, Wen-Rong; Tian, Bo; Jiang, Yan; Zhen, Hui-Ling

2015-02-01

Optical rogue waves of the coupled nonlinear Schrödinger equations with negative coherent coupling, which describe the propagation of orthogonally polarized optical waves in an isotropic medium, are reported. We construct and discuss a family of the vector rogue-wave solutions, including the bright rogue waves, four-petaled rogue waves, and dark rogue waves. A bright rogue wave without a valley can split up, giving birth to two bright rogue waves, and an eye-shaped rogue wave can split up, giving birth to two dark rogue waves.

Finite Difference Modeling of Wave Progpagation in Acoustic TiltedTI Media

DOE Office of Scientific and Technical Information (OSTI.GOV)

Zhang, Linbin; Rector III, James W.; Hoversten, G. Michael

2005-03-21

Based on an acoustic assumption (shear wave velocity is zero) and a dispersion relation, we derive an acoustic wave equation for P-waves in tilted transversely isotropic (TTI) media (transversely isotropic media with a tilted symmetry axis). This equation has fewer parameters than an elastic wave equation in TTI media and yields an accurate description of P-wave traveltimes and spreading-related attenuation. Our TTI acoustic wave equation is a fourth-order equation in time and space. We demonstrate that the acoustic approximation allows the presence of shear waves in the solution. The substantial differences in traveltime and amplitude between data created using VTImore » and TTI assumptions is illustrated in examples.« less

Contribution of non-resonant wave-wave interactions in the dynamics of long-crested sea wave fields

NASA Astrophysics Data System (ADS)

Benoit, Michel

2017-04-01

Gravity waves fields at the surface of the oceans evolve under the combined effects of several physical mechanisms, of which nonlinear wave-wave interactions play a dominant role. These interactions transfer energy between components within the energy spectrum and allow in particular to explain the shape of the distribution of wave energy according to the frequencies and directions of propagation. In the oceanic domain (deep water conditions), dominant interactions are third-order resonant interactions, between quadruplets (or quartets) of wave components, and the evolution of the wave spectrum is governed by a kinetic equation, established by Hasselmann (1962) and Zakharov (1968). The kinetic equation has a number of interesting properties, including the existence of self-similar solutions and cascades to small and large wavelengths of waves, which can be studied in the framework of the wave (or weak) turbulence theory (e.g. Badulin et al., 2005). With the aim to obtain more complete and precise modelling of sea states dynamics, we investigate here the possibility and consequences of taking into account the non-resonant interactions -quasi-resonant in practice- among 4 waves. A mathematical formalism has recently been proposed to account for these non-resonant interactions in a statistical framework by Annenkov & Shrira (2006) (Generalized Kinetic Equation, GKE) and Gramstad & Stiassnie (2013) (Phase Averaged Equation, PAE). In order to isolate the non-resonant contributions, we limit ourselves here to monodirectional (i.e. long-crested) wave trains, since in this case the 4-wave resonant interactions vanish. The (stochastic) modelling approaches proposed by Annenkov & Shrira (2006) and Gramstad & Stiassnie (2013) are compared to phase-resolving (deterministic) simulations based on a fully nonlinear potential approach (using a high-order spectral method, HOS). We study and compare the evolution dynamics of the wave spectrum at different time scales (i.e. over durations ranging from a few wave periods to 1000 periods), with the aim of highlighting the capabilities and limitations of the GKE-PAE models. Different situations are considered by varying the relative water depth, the initial steepness of the wave field, and the shape of the initial wave spectrum, including arbitrary forms. References: Annenkov S.Y., Shrira V.I. (2006) Role of non-resonant interactions in the evolution of nonlinear random water wave fields. J. Fluid Mech., 561, 181-207. Badulin S.I., Pushkarev A.N., Resio D., Zakharov V.E. (2005) Self-similarity of wind-driven seas. Nonlin. Proc. Geophys., 12, 891-946. Gramstad O., Stiassnie M. (2013) Phase-averaged equation for water waves. J. Fluid Mech., 718, 280- 303. Hasselmann K. (1962) On the non-linear energy transfer in a gravity-wave spectrum. Part 1. General theory. J. Fluid Mech., 12, 481-500. Zakharov V.E. (1968) Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. App. Mech. Tech. Phys., 9(2), 190-194.

True amplitude wave equation migration arising from true amplitude one-way wave equations

NASA Astrophysics Data System (ADS)

Zhang, Yu; Zhang, Guanquan; Bleistein, Norman

2003-10-01

One-way wave operators are powerful tools for use in forward modelling and inversion. Their implementation, however, involves introduction of the square root of an operator as a pseudo-differential operator. Furthermore, a simple factoring of the wave operator produces one-way wave equations that yield the same travel times as the full wave equation, but do not yield accurate amplitudes except for homogeneous media and for almost all points in heterogeneous media. Here, we present augmented one-way wave equations. We show that these equations yield solutions for which the leading order asymptotic amplitude as well as the travel time satisfy the same differential equations as the corresponding functions for the full wave equation. Exact representations of the square-root operator appearing in these differential equations are elusive, except in cases in which the heterogeneity of the medium is independent of the transverse spatial variables. Here, we address the fully heterogeneous case. Singling out depth as the preferred direction of propagation, we introduce a representation of the square-root operator as an integral in which a rational function of the transverse Laplacian appears in the integrand. This allows us to carry out explicit asymptotic analysis of the resulting one-way wave equations. To do this, we introduce an auxiliary function that satisfies a lower dimensional wave equation in transverse spatial variables only. We prove that ray theory for these one-way wave equations leads to one-way eikonal equations and the correct leading order transport equation for the full wave equation. We then introduce appropriate boundary conditions at z = 0 to generate waves at depth whose quotient leads to a reflector map and an estimate of the ray theoretical reflection coefficient on the reflector. Thus, these true amplitude one-way wave equations lead to a 'true amplitude wave equation migration' (WEM) method. In fact, we prove that applying the WEM imaging condition to these newly defined wavefields in heterogeneous media leads to the Kirchhoff inversion formula for common-shot data when the one-way wavefields are replaced by their ray theoretic approximations. This extension enhances the original WEM method. The objective of that technique was a reflector map, only. The underlying theory did not address amplitude issues. Computer output obtained using numerically generated data confirms the accuracy of this inversion method. However, there are practical limitations. The observed data must be a solution of the wave equation. Therefore, the data over the entire survey area must be collected from a single common-shot experiment. Multi-experiment data, such as common-offset data, cannot be used with this method as currently formulated. Research on extending the method is ongoing at this time.

Propagation estimates for dispersive wave equations: Application to the stratified wave equation

NASA Astrophysics Data System (ADS)

Pravica, David W.

1999-01-01

The plane-stratified wave equation (∂t2+H)ψ=0 with H=-c(y)2∇z2 is studied, where z=x⊕y, x∈Rk, y∈R1 and |c(y)-c∞|→0 as |y|→∞. Solutions to such an equation are solved for the propagation of waves through a layered medium and can include waves which propagate in the x-directions only (i.e., trapped modes). This leads to a consideration of the pseudo-differential wave equation (∂t2+ω(-Δx))ψ=0 such that the dispersion relation ω(ξ2) is analytic and satisfies c1⩽ω'(ξ2)⩽c2 for c*>0. Uniform propagation estimates like ∫|x|⩽|t|αE(UtP±φ0)dkx⩽Cα,β(1+|t|)-β∫E(φ0)dkx are obtained where Ut is the evolution group, P± are projection operators onto the Hilbert space of initial conditions φ∈H and E(ṡ) is the local energy density. In special cases scattering of trapped modes off a local perturbation satisfies the causality estimate ||P+ρΛjSP-ρΛk||⩽Cνρ-ν for each ν<1/2. Here P+ρΛj (P-ρΛk) are remote outgoing/detector (incoming/transmitter) projections for the jth (kth) trapped mode. Also Λ⋐R+ is compact, so the projections localize onto formally-incoming (eventually-outgoing) states.

A Numerical Investigation of the Burnett Equations Based on the Second Law

NASA Technical Reports Server (NTRS)

Comeaux, Keith A.; Chapman, Dean R.; MacCormack, Robert W.; Edwards, Thomas A. (Technical Monitor)

1995-01-01

The Burnett equations have been shown to potentially violate the second law of thermodynamics. The objective of this investigation is to correlate the numerical problems experienced by the Burnett equations to the negative production of entropy. The equations have had a long history of numerical instability to small wavelength disturbances. Recently, Zhong corrected the instability problem and made solutions attainable for one dimensional shock waves and hypersonic blunt bodies. Difficulties still exist when attempting to solve hypersonic flat plate boundary layers and blunt body wake flows, however. Numerical experiments will include one-dimensional shock waves, quasi-one dimensional nozzles, and expanding Prandlt-Meyer flows and specifically examine the entropy production for these cases.

Optical soliton solutions, periodic wave solutions and complexitons of the cubic Schrödinger equation with a bounded potential

NASA Astrophysics Data System (ADS)

Yan, Xue-Wei; Tian, Shou-Fu; Dong, Min-Jie; Zou, Li

2018-01-01

In this paper, we consider the cubic Schrödinger equation with a bounded potential, which describes the propagation properties of optical soliton solutions. By employing an ansatz method, we precisely derive the bright and dark soliton solutions of the equation. Moreover, we obtain three classes of analytic periodic wave solutions expressed in terms of the Jacobi's elliptic functions including cn ,sn and dn functions. Finally, by using a tanh function method, its complexitons solutions are derived in a very natural way. It is hoped that our results can enrich the nonlinear dynamical behaviors of the cubic Schrödinger equation with a bounded potential.

Bright, dark and W-shaped solitons with extended nonlinear Schrödinger's equation for odd and even higher-order terms

NASA Astrophysics Data System (ADS)

Bendahmane, Issam; Triki, Houria; Biswas, Anjan; Saleh Alshomrani, Ali; Zhou, Qin; Moshokoa, Seithuti P.; Belic, Milivoj

2018-02-01

We present solitary wave solutions of an extended nonlinear Schrödinger equation with higher-order odd (third-order) and even (fourth-order) terms by using an ansatz method. The including high-order dispersion terms have significant physical applications in fiber optics, the Heisenberg spin chain, and ocean waves. Exact envelope solutions comprise bright, dark and W-shaped solitary waves, illustrating the potentially rich set of solitary wave solutions of the extended model. Furthermore, we investigate the properties of these solitary waves in nonlinear and dispersive media. Moreover, specific constraints on the system parameters for the existence of these structures are discussed exactly. The results show that the higher-order dispersion and nonlinear effects play a crucial role for the formation and properties of propagating waves.

A Finite-Difference Time-Domain Model of Artificial Ionospheric Modification

NASA Astrophysics Data System (ADS)

Cannon, Patrick; Honary, Farideh; Borisov, Nikolay

Experiments in the artificial modification of the ionosphere via a radio frequency pump wave have observed a wide range of non-linear phenomena near the reflection height of an O-mode wave. These effects exhibit a strong aspect-angle dependence thought to be associated with the process by which, for a narrow range of off-vertical launch angles, the O-mode pump wave can propagate beyond the standard reflection height at X=1 as a Z-mode wave and excite additional plasma activity. A numerical model based on Finite-Difference Time-Domain method has been developed to simulate the interaction of the pump wave with an ionospheric plasma and investigate different non-linear processes involved in modification experiments. The effects on wave propagation due to plasma inhomogeneity and anisotropy are introduced through coupling of the Lorentz equation of motion for electrons and ions to Maxwell’s wave equations in the FDTD formulation, leading to a model that is capable of exciting a variety of plasma waves including Langmuir and upper-hybrid waves. Additionally, discretized equations describing the time-dependent evolution of the plasma fluid temperature and density are included in the FDTD update scheme. This model is used to calculate the aspect angle dependence and angular size of the radio window for which Z-mode excitation occurs, and the results compared favourably with both theoretical predictions and experimental observations. The simulation results are found to reproduce the angular dependence on electron density and temperature enhancement observed experimentally. The model is used to investigate the effect of different initial plasma density conditions on the evolution of non-linear effects, and demonstrates that the inclusion of features such as small field-aligned density perturbations can have a significant influence on wave propagation and the magnitude of temperature and density enhancements.

Self-Consistent Model of Magnetospheric Ring Current and Propagating Electromagnetic Ion Cyclotron Waves: Waves in Multi-Ion Magnetosphere

NASA Technical Reports Server (NTRS)

Khazanov, G. V.; Gamayunov, K. V.; Gallagher, D. L.; Kozyra, J. U.

2006-01-01

The further development of a self-consistent theoretical model of interacting ring current ions and electromagnetic ion cyclotron waves (Khazanov et al., 2003) is presented In order to adequately take into account wave propagation and refraction in a multi-ion magnetosphere, we explicitly include the ray tracing equations in our previous self-consistent model and use the general form of the wave kinetic equation. This is a major new feature of the present model and, to the best of our knowledge, the ray tracing equations for the first time are explicitly employed on a global magnetospheric scale in order to self-consistently simulate the spatial, temporal, and spectral evolution of the ring current and of electromagnetic ion cyclotron waves To demonstrate the effects of EMIC wave propagation and refraction on the wave energy distribution and evolution, we simulate the May 1998 storm. The main findings of our simulation can be summarized as follows. First, owing to the density gradient at the plasmapause, the net wave refraction is suppressed, and He+-mode grows preferably at the plasmapause. This result is in total agreement with previous ray tracing studies and is very clearly found in presented B field spectrograms. Second, comparison of global wave distributions with the results from another ring current model (Kozyra et al., 1997) reveals that this new model provides more intense and more highly plasmapause-organized wave distributions during the May 1998 storm period Finally, it is found that He(+)-mode energy distributions are not Gaussian distributions and most important that wave energy can occupy not only the region of generation, i.e., the region of small wave normal angles, but all wave normal angles, including those to near 90 . The latter is extremely crucial for energy transfer to thermal plasmaspheric electrons by resonant Landau damping and subsequent downward heat transport and excitation of stable auroral red arcs.

Modeling unsteady sound refraction by coherent structures in a high-speed jet

NASA Astrophysics Data System (ADS)

Kan, Pinqing; Lewalle, Jacques

2011-11-01

We construct a visual model for the unsteady refraction of sound waves from point sources in a Ma = 0.6 jet. The mass and inviscid momentum equations give an equation governing acoustic fluctuations, including anisotropic propagation, attenuation and sources; differences with Lighthill's equation will be discussed. On this basis, the theory of characteristics gives canonical equations for the acoustic paths from any source into the far field. We model a steady mean flow in the near-jet region including the potential core and the mixing region downstream of its collapse, and model the convection of coherent structures as traveling wave perturbations of this mean flow. For a regular distribution of point sources in this region, we present a visual rendition of fluctuating distortion, lensing and deaf spots from the viewpoint of a far-field observer. Supported in part by AFOSR Grant FA-9550-10-1-0536 and by a Syracuse University Graduate Fellowship.

Exact traveling wave solutions of modified KdV-Zakharov-Kuznetsov equation and viscous Burgers equation.

PubMed

Islam, Md Hamidul; Khan, Kamruzzaman; Akbar, M Ali; Salam, Md Abdus

2014-01-01

Mathematical modeling of many physical systems leads to nonlinear evolution equations because most physical systems are inherently nonlinear in nature. The investigation of traveling wave solutions of nonlinear partial differential equations (NPDEs) plays a significant role in the study of nonlinear physical phenomena. In this article, we construct the traveling wave solutions of modified KDV-ZK equation and viscous Burgers equation by using an enhanced (G '/G) -expansion method. A number of traveling wave solutions in terms of unknown parameters are obtained. Derived traveling wave solutions exhibit solitary waves when special values are given to its unknown parameters. 35C07; 35C08; 35P99.

An approach to rogue waves through the cnoidal equation

NASA Astrophysics Data System (ADS)

Lechuga, Antonio

2014-05-01

Lately it has been realized the importance of rogue waves in some events happening in open seas. Extreme waves and extreme weather could explain some accidents, but not all of them. Every now and then inflicted damages on ships only can be reported to be caused by anomalous and elusive waves, such as rogue waves. That's one of the reason why they continue attracting considerable interest among researchers. In the frame of the Nonlinear Schrödinger equation(NLS), Witham(1974) and Dingemans and Otta (2001)gave asymptotic solutions in moving coordinates that transformed the NLS equation in a ordinary differential equation that is the Duffing or cnoidal wave equation. Applying the Zakharov equation, Stiassnie and Shemer(2004) and Shemer(2010)got also a similar equation. It's well known that this ordinary equation can be solved in elliptic functions. The main aim of this presentation is to sort out the domains of the solutions of this equation, that, of course, are linked to the corresponding solutions of the partial differential equations(PDEs). That being, Lechuga(2007),a simple way to look for anomalous waves as it's the case with some "chaotic" solutions of the Duffing equation.

Wave equations in conformal gravity

NASA Astrophysics Data System (ADS)

Du, Juan-Juan; Wang, Xue-Jing; He, You-Biao; Yang, Si-Jiang; Li, Zhong-Heng

2018-05-01

We study the wave equation governing massless fields of all spins (s = 0, 1 2, 1, 3 2 and 2) in the most general spherical symmetric metric of conformal gravity. The equation is separable, the solution of the angular part is a spin-weighted spherical harmonic, and the radial wave function may be expressed in terms of solutions of the Heun equation which has four regular singular points. We also consider various special cases of the metric and find that the angular wave functions are the same for all cases, the actual shape of the metric functions affects only the radial wave function. It is interesting to note that each radial equation can be transformed into a known ordinary differential equation (i.e. Heun equation, or confluent Heun equation, or hypergeometric equation). The results show that there are analytic solutions for all the wave equations of massless spin fields in the spacetimes of conformal gravity. This is amazing because exact solutions are few and far between for other spacetimes.

Exact traveling wave solutions of fractional order Boussinesq-like equations by applying Exp-function method

NASA Astrophysics Data System (ADS)

Rahmatullah; Ellahi, Rahmat; Mohyud-Din, Syed Tauseef; Khan, Umar

2018-03-01

We have computed new exact traveling wave solutions, including complex solutions of fractional order Boussinesq-Like equations, occurring in physical sciences and engineering, by applying Exp-function method. The method is blended with fractional complex transformation and modified Riemann-Liouville fractional order operator. Our obtained solutions are verified by substituting back into their corresponding equations. To the best of our knowledge, no other technique has been reported to cope with the said fractional order nonlinear problems combined with variety of exact solutions. Graphically, fractional order solution curves are shown to be strongly related to each other and most importantly, tend to fixate on their integer order solution curve. Our solutions comprise high frequencies and very small amplitude of the wave responses.

New classes of solutions in the coupled PT symmetric nonlocal nonlinear Schrödinger equations with four wave mixing

NASA Astrophysics Data System (ADS)

Vinayagam, P. S.; Radha, R.; Al Khawaja, U.; Ling, Liming

2018-06-01

We investigate generalized nonlocal coupled nonlinear Schorödinger equation containing Self-Phase Modulation, Cross-Phase Modulation and four wave mixing involving nonlocal interaction. By means of Darboux transformation we obtained a family of exact breathers and solitons including the Peregrine soliton, Kuznetsov-Ma breather, Akhmediev breather along with all kinds of soliton-soliton and breather-soltion interactions. We analyze and emphasize the impact of the four-wave mixing on the nature and interaction of the solutions. We found that the presence of four wave mixing converts a two-soliton solution into an Akhmediev breather. In particular, the inclusion of four wave mixing results in the generation of a new solutions which is spatially and temporally periodic called "Soliton (Breather) lattice".

DOE Office of Scientific and Technical Information (OSTI.GOV)

Andreev, Pavel A., E-mail: andreevpa@physics.msu.ru; Kuz’menkov, L.S., E-mail: lsk@phys.msu.ru

We consider quantum plasmas of electrons and motionless ions. We describe separate evolution of spin-up and spin-down electrons. We present corresponding set of quantum hydrodynamic equations. We assume that plasmas are placed in an uniform external magnetic field. We account different occupation of spin-up and spin-down quantum states in equilibrium degenerate plasmas. This effect is included via equations of state for pressure of each species of electrons. We study oblique propagation of longitudinal waves. We show that instead of two well-known waves (the Langmuir wave and the Trivelpiece–Gould wave), plasmas reveal four wave solutions. New solutions exist due to bothmore » the separate consideration of spin-up and spin-down electrons and different occupation of spin-up and spin-down quantum states in equilibrium state of degenerate plasmas.« less

Rogue periodic waves of the modified KdV equation

NASA Astrophysics Data System (ADS)

Chen, Jinbing; Pelinovsky, Dmitry E.

2018-05-01

Rogue periodic waves stand for rogue waves on a periodic background. Two families of travelling periodic waves of the modified Korteweg–de Vries (mKdV) equation in the focusing case are expressed by the Jacobian elliptic functions dn and cn. By using one-fold and two-fold Darboux transformations of the travelling periodic waves, we construct new explicit solutions for the mKdV equation. Since the dn-periodic wave is modulationally stable with respect to long-wave perturbations, the new solution constructed from the dn-periodic wave is a nonlinear superposition of an algebraically decaying soliton and the dn-periodic wave. On the other hand, since the cn-periodic wave is modulationally unstable with respect to long-wave perturbations, the new solution constructed from the cn-periodic wave is a rogue wave on the cn-periodic background, which generalizes the classical rogue wave (the so-called Peregrine’s breather) of the nonlinear Schrödinger equation. We compute the magnification factor for the rogue cn-periodic wave of the mKdV equation and show that it remains constant for all amplitudes. As a by-product of our work, we find explicit expressions for the periodic eigenfunctions of the spectral problem associated with the dn and cn periodic waves of the mKdV equation.

Reply to "Comment on 'A Self-Consistent Model of the Interacting Ring Current Ions and Electromagnetic Ion Cyclotron Waves, Initial Results: Waves and Precipitation Fluxes' and 'Self-Consistent Model of the Magnetospheric Ring Current and Propagating Electromagnetic Ion Cyclotron Waves: Waves in Multi-Ion Magnetosphere' by Khazanov et al. et al."

NASA Technical Reports Server (NTRS)

Khazanov, G. V.; Gamayunov, K. V.; Gallagher, D. L.; Kozyra, J. W.

2007-01-01

It is well-known that the effects of electromagnetic ion cyclotron (EMIC) waves on ring current (RC) ion and radiation belt (RB) electron dynamics strongly depend on such particle/wave characteristics as the phase-space distribution function, frequency, wavenormal angle, wave energy, and the form of wave spectral energy density. The consequence is that accurate modeling of EMIC waves and RC particles requires robust inclusion of the interdependent dynamics of wave growth/damping, wave propagation, and[ particles. Such a self-consistent model is being progressively developed by Khazanov et al. [2002, 2006, 2007]. This model is based on a system of coupled kinetic equations for the RC and EMIC wave power spectral density along with the ray tracing equations. Thome and Home [2007] (hereafter referred to as TH2007) call the Khazanov et al. [2002, 2006] results into question in their Comment. The points in contention can be summarized as follows. TH2007 claim that: (1) "the important damping of waves by thermal heavy ions is completely ignored", and Landau damping during resonant interaction with thermal electrons is not included in our model; (2) EMIC wave damping due to RC O + is not included in our simulation; (3) non-linear processes limiting EMIC wave amplitude are not included in our model; (4) growth of the background fluctuations to a physically significantamplitude"must occur during a single transit of the unstable region" with subsequent damping below bi-ion latitudes,and consequently"the bounce averaged wave kinetic equation employed in the code contains a physically erroneous 'assumption". Our reply will address each of these points as well as other criticisms mentioned in the Comment. TH2007 are focused on two of our papers that are separated by four years. Significant progress in the self-consistent treatment of the RC-EMIC wave system has been achieved during those years. The paper by Khazanov et al. [2006] presents the latest version of our model, and in this Reply we refer mostly to this paper.

Second-harmonic generation in shear wave beams with different polarizations

NASA Astrophysics Data System (ADS)

Spratt, Kyle S.; Ilinskii, Yurii A.; Zabolotskaya, Evgenia A.; Hamilton, Mark F.

2015-10-01

A coupled pair of nonlinear parabolic equations was derived by Zabolotskaya [1] that model the transverse components of the particle motion in a collimated shear wave beam propagating in an isotropic elastic solid. Like the KZK equation, the parabolic equation for shear wave beams accounts consistently for the leading order effects of diffraction, viscosity and nonlinearity. The nonlinearity includes a cubic nonlinear term that is equivalent to that present in plane shear waves, as well as a quadratic nonlinear term that is unique to diffracting beams. The work by Wochner et al. [2] considered shear wave beams with translational polarizations (linear, circular and elliptical), wherein second-order nonlinear effects vanish and the leading order nonlinear effect is third-harmonic generation by the cubic nonlinearity. The purpose of the current work is to investigate the quadratic nonlinear term present in the parabolic equation for shear wave beams by considering second-harmonic generation in Gaussian beams as a second-order nonlinear effect using standard perturbation theory. In order for second-order nonlinear effects to be present, a broader class of source polarizations must be considered that includes not only the familiar translational polarizations, but also polarizations accounting for stretching, shearing and rotation of the source plane. It is found that the polarization of the second harmonic generated by the quadratic nonlinearity is not necessarily the same as the polarization of the source-frequency beam, and we are able to derive a general analytic solution for second-harmonic generation from a Gaussian source condition that gives explicitly the relationship between the polarization of the source-frequency beam and the polarization of the second harmonic.

Numerical simulation of electromagnetic wave attenuation in a nonequilibrium chemically reacting hypervelocity flow

NASA Astrophysics Data System (ADS)

Nusca, Michael Joseph, Jr.

The effects of various gasdynamic phenomena on the attenuation of an electromagnetic wave propagating through the nonequilibrium chemically reacting air flow field generated by an aerodynamic body travelling at high velocity is investigated. The nonequilibrium flow field is assumed to consist of seven species including nitric oxide ions and free electrons. The ionization of oxygen and nitrogen atoms is ignored. The aerodynamic body considered is a blunt wedge. The nonequilibrium chemically reacting flow field around this body is numerically simulated using a computer code based on computational fluid dynamics. The computer code solves the Navier-Stokes equations including mass diffusion and heat transfer, using a time-marching, explicit Runge-Kutta scheme. A nonequilibrium air kinetics model consisting of seven species and twenty-eight reactions as well as an equilibrium air model consisting of the same seven species are used. The body surface boundaries are considered as adiabatic or isothermal walls, as well as fully-catalytic and non-catalytic surfaces. Both laminar and turbulent flows are considered; wall generated flow turbulence is simulated using an algebraic mixing length model. An electromagnetic wave is considered as originating from an antenna within the body and is effected by the free electrons in the chemically reacting flow. Analysis of the electromagnetics is performed separately from the fluid dynamic analysis using a series solution of Maxwell's equations valid for the propagation of a long-wavelength plane electromagnetic wave through a thin (i.e., in comparison to wavelength) inhomogeneous plasma layer. The plasma layer is the chemically reacting shock layer around the body. The Navier-Stokes equations are uncoupled from Maxwell's equations. The results of this computational study demonstrate for the first time and in a systematic fashion, the importance of several parameters including equilibrium chemistry, nonequilibrium chemical kinetics, the reaction mechanism, flow viscosity, mass diffusion, and wall boundary conditions on modeling wave attenuation resulting from the interaction of an electromagnetic wave with an aerodynamic plasma. Comparison is made with experimental data.

Wave-packet formation at the zero-dispersion point in the Gardner-Ostrovsky equation.

PubMed

Whitfield, A J; Johnson, E R

2015-05-01

The long-time effect of weak rotation on an internal solitary wave is the decay into inertia-gravity waves and the eventual emergence of a coherent, steadily propagating, nonlinear wave packet. There is currently no entirely satisfactory explanation as to why these wave packets form. Here the initial value problem is considered within the context of the Gardner-Ostrovsky, or rotation-modified extended Korteweg-de Vries, equation. The linear Gardner-Ostrovsky equation has maximum group velocity at a critical wave number, often called the zero-dispersion point. It is found here that a nonlinear splitting of the wave-number spectrum at the zero-dispersion point, where energy is shifted into the modulationally unstable regime of the Gardner-Ostrovsky equation, is responsible for the wave-packet formation. Numerical comparisons of the decay of a solitary wave in the Gardner-Ostrovsky equation and a derived nonlinear Schrödinger equation at the zero-dispersion point are used to confirm the spectral splitting.

Convective wave breaking in the KdV equation

NASA Astrophysics Data System (ADS)

Brun, Mats K.; Kalisch, Henrik

2018-03-01

The KdV equation is a model equation for waves at the surface of an inviscid incompressible fluid, and it is well known that the equation describes the evolution of unidirectional waves of small amplitude and long wavelength fairly accurately if the waves fall into the Boussinesq regime. The KdV equation allows a balance of nonlinear steepening effects and dispersive spreading which leads to the formation of steady wave profiles in the form of solitary waves and cnoidal waves. While these wave profiles are solutions of the KdV equation for any amplitude, it is shown here that there for both the solitary and the cnoidal waves, there are critical amplitudes for which the horizontal component of the particle velocity matches the phase velocity of the wave. Solitary or cnoidal solutions of the KdV equation which surpass these amplitudes feature incipient wave breaking as the particle velocity exceeds the phase velocity near the crest of the wave, and the model breaks down due to violation of the kinematic surface boundary condition. The condition for breaking can be conveniently formulated as a convective breaking criterion based on the local Froude number at the wave crest. This breaking criterion can also be applied to time-dependent situations, and one case of interest is the development of an undular bore created by an influx at a lateral boundary. It is shown that this boundary forcing leads to wave breaking in the leading wave behind the bore if a certain threshold is surpassed.

Wave propagation problem for a micropolar elastic waveguide

NASA Astrophysics Data System (ADS)

Kovalev, V. A.; Murashkin, E. V.; Radayev, Y. N.

2018-04-01

A propagation problem for coupled harmonic waves of translational displacements and microrotations along the axis of a long cylindrical waveguide is discussed at present study. Microrotations modeling is carried out within the linear micropolar elasticity frameworks. The mathematical model of the linear (or even nonlinear) micropolar elasticity is also expanded to a field theory model by variational least action integral and the least action principle. The governing coupled vector differential equations of the linear micropolar elasticity are given. The translational displacements and microrotations in the harmonic coupled wave are decomposed into potential and vortex parts. Calibrating equations providing simplification of the equations for the wave potentials are proposed. The coupled differential equations are then reduced to uncoupled ones and finally to the Helmholtz wave equations. The wave equations solutions for the translational and microrotational waves potentials are obtained for a high-frequency range.

Symmetry Reductions and Group-Invariant Radial Solutions to the n-Dimensional Wave Equation

NASA Astrophysics Data System (ADS)

Feng, Wei; Zhao, Songlin

2018-01-01

In this paper, we derive explicit group-invariant radial solutions to a class of wave equation via symmetry group method. The optimal systems of one-dimensional subalgebras for the corresponding radial wave equation are presented in terms of the known point symmetries. The reductions of the radial wave equation into second-order ordinary differential equations (ODEs) with respect to each symmetry in the optimal systems are shown. Then we solve the corresponding reduced ODEs explicitly in order to write out the group-invariant radial solutions for the wave equation. Finally, several analytical behaviours and smoothness of the resulting solutions are discussed.

Diffraction of Harmonic Flexural Waves in a Cracked Elastic Plate Carrying Electrical Current

NASA Technical Reports Server (NTRS)

Ambur, Damodar R.; Hasanyan, Davresh; Librescu, iviu; Qin, Zhanming

2005-01-01

The scattering effect of harmonic flexural waves at a through crack in an elastic plate carrying electrical current is investigated. In this context, the Kirchhoffean bending plate theory is extended as to include magnetoelastic interactions. An incident wave giving rise to bending moments symmetric about the longitudinal z-axis of the crack is applied. Fourier transform technique reduces the problem to dual integral equations, which are then cast to a system of two singular integral equations. Efficient numerical computation is implemented to get the bending moment intensity factor for arbitrary frequency of the incident wave and of arbitrary electrical current intensity. The asymptotic behaviour of the bending moment intensity factor is analysed and parametric studies are conducted.

Numerical Simulations of Light Bullets, Using The Full Vector, Time Dependent, Nonlinear Maxwell Equations

NASA Technical Reports Server (NTRS)

Goorjian, Peter M.; Silberberg, Yaron; Kwak, Dochan (Technical Monitor)

1994-01-01

This paper will present results in computational nonlinear optics. An algorithm will be described that solves the full vector nonlinear Maxwell's equations exactly without the approximations that are currently made. Present methods solve a reduced scalar wave equation, namely the nonlinear Schrodinger equation, and neglect the optical carrier. Also, results will be shown of calculations of 2-D electromagnetic nonlinear waves computed by directly integrating in time the nonlinear vector Maxwell's equations. The results will include simulations of 'light bullet' like pulses. Here diffraction and dispersion will be counteracted by nonlinear effects. The time integration efficiently implements linear and nonlinear convolutions for the electric polarization, and can take into account such quantum effects as Kerr and Raman interactions. The present approach is robust and should permit modeling 2-D and 3-D optical soliton propagation, scattering, and switching directly from the full-vector Maxwell's equations.

Numerical Simulations of Light Bullets, Using The Full Vector, Time Dependent, Nonlinear Maxwell Equations

NASA Technical Reports Server (NTRS)

Goorjian, Peter M.; Silberberg, Yaron; Kwak, Dochan (Technical Monitor)

1995-01-01

This paper will present results in computational nonlinear optics. An algorithm will be described that solves the full vector nonlinear Maxwell's equations exactly without the approximations that we currently made. Present methods solve a reduced scalar wave equation, namely the nonlinear Schrodinger equation, and neglect the optical carrier. Also, results will be shown of calculations of 2-D electromagnetic nonlinear waves computed by directly integrating in time the nonlinear vector Maxwell's equations. The results will include simulations of 'light bullet' like pulses. Here diffraction and dispersion will be counteracted by nonlinear effects. The time integration efficiently implements linear and nonlinear convolutions for the electric polarization, and can take into account such quantum effects as Karr and Raman interactions. The present approach is robust and should permit modeling 2-D and 3-D optical soliton propagation, scattering, and switching directly from the full-vector Maxwell's equations.

Periodic wave, breather wave and travelling wave solutions of a (2 + 1)-dimensional B-type Kadomtsev-Petviashvili equation in fluids or plasmas

NASA Astrophysics Data System (ADS)

Hu, Wen-Qiang; Gao, Yi-Tian; Jia, Shu-Liang; Huang, Qian-Min; Lan, Zhong-Zhou

2016-11-01

In this paper, a (2 + 1)-dimensional B-type Kadomtsev-Petviashvili equation is investigated, which has been presented as a model for the shallow water wave in fluids or the electrostatic wave potential in plasmas. By virtue of the binary Bell polynomials, the bilinear form of this equation is obtained. With the aid of the bilinear form, N -soliton solutions are obtained by the Hirota method, periodic wave solutions are constructed via the Riemann theta function, and breather wave solutions are obtained according to the extended homoclinic test approach. Travelling waves are constructed by the polynomial expansion method as well. Then, the relations between soliton solutions and periodic wave solutions are strictly established, which implies the asymptotic behaviors of the periodic waves under a limited procedure. Furthermore, we obtain some new solutions of this equation by the standard extended homoclinic test approach. Finally, we give a generalized form of this equation, and find that similar analytical solutions can be obtained from the generalized equation with arbitrary coefficients.

Lie symmetry analysis, conservation laws, solitary and periodic waves for a coupled Burger equation

NASA Astrophysics Data System (ADS)

Xu, Mei-Juan; Tian, Shou-Fu; Tu, Jian-Min; Zhang, Tian-Tian

2017-01-01

Under investigation in this paper is a generalized (2 + 1)-dimensional coupled Burger equation with variable coefficients, which describes lots of nonlinear physical phenomena in geophysical fluid dynamics, condense matter physics and lattice dynamics. By employing the Lie group method, the symmetry reductions and exact explicit solutions are obtained, respectively. Based on a direct method, the conservations laws of the equation are also derived. Furthermore, by virtue of the Painlevé analysis, we successfully obtain the integrable condition on the variable coefficients, which plays an important role in further studying the integrability of the equation. Finally, its auto-Bäcklund transformation as well as some new analytic solutions including solitary and periodic waves are also presented via algebraic and differential manipulation.

On nonlinear evolution of low-frequency Alfvén waves in weakly-expanding solar wind plasmas

DOE Office of Scientific and Technical Information (OSTI.GOV)

Nariyuki, Y.

A multi-dimensional nonlinear evolution equation for Alfvén waves in weakly-expanding solar wind plasmas is derived by using the reductive perturbation method. The expansion of solar wind plasma parcels is modeled by an expanding box model, which includes the accelerating expansion. It is shown that the resultant equation agrees with the Wentzel-Kramers-Brillouin prediction of the low-frequency Alfvén waves in the linear limit. In the cold and one-dimensional limit, a modified derivative nonlinear Schrodinger equation is obtained. Direct numerical simulations are carried out to discuss the effect of the expansion on the modulational instability of monochromatic Alfvén waves and the propagation ofmore » Alfvén solitons. By using the instantaneous frequency, it is quantitatively shown that as far as the expansion rate is much smaller than wave frequencies, effects of the expansion are almost adiabatic. It is also confirmed that while shapes of Alfvén solitons temporally change due to the expansion, some of them can stably propagate after their collision in weakly-expanding plasmas.« less

Wave and plasma observations during a compressional Pc 5 wave event August 10, 1982

NASA Technical Reports Server (NTRS)

Engebretson, M. J.; Cahill, L. J., Jr.; Waite, J. H., Jr.; Gallagher, D. L.; Chandler, M. O.; Sugiura, M.

1986-01-01

Magnetometer and thermal plasma instruments on the polar-orbiting Dynamics Explorer 1 satellite observed a small-amplitude ultralow frequency pulsation event at the outer edge of the plasmapause near the geomagnetic equator in the midafternoon sector on August 10, 1982, during the recovery phase of a magnetic storm. Transverse pulsations of 30-50 s period were observed throughout the event, and a 270-s period, purely compressional Pc 5 pulsation with several shifts in phase occurred within + or - 5 deg of the geomagnetic equator. Electric fields and the motion of thermal ions appeared to be in quadrature with pulsations in magnetic field magnitude throughout the event. This suggests that the net Poynting flux for the compressional waves was zero, consistent with their being standing waves. Large fluxes of trapped 90 deg pitch angle 10-eV protons, also symmetric about the geomagnetic equator, were observed in conjunction with the waves. These may serve as a source of free energy for the pulsations. These observations lend support to recent studies suggesting that many dayside compressional wave events are related to localized field line resonance near plasmapauselike boundaries, but also include features that cannot be explained by existing theories.

Wave and plasma observations during a compressional Pc 5 wave event August 10, 1982

NASA Astrophysics Data System (ADS)

Engebretson, M. J.; Cahill, L. J., Jr.; Waite, J. H., Jr.; Gallagher, D. L.; Chandler, M. O.; Sugiura, M.; Weimer, D. R.

1986-06-01

Magnetometer and thermal plasma instruments on the polar-orbiting Dynamics Explorer 1 satellite observed a small-amplitude ultralow frequency pulsation event at the outer edge of the plasmapause near the geomagnetic equator in the midafternoon sector on August 10, 1982, during the recovery phase of a magnetic storm. Transverse pulsations of 30-50 s period were observed throughout the event, and a 270-s period, purely compressional Pc 5 pulsation with several shifts in phase occurred within + or - 5 deg of the geomagnetic equator. Electric fields and the motion of thermal ions appeared to be in quadrature with pulsations in magnetic field magnitude throughout the event. This suggests that the net Poynting flux for the compressional waves was zero, consistent with their being standing waves. Large fluxes of trapped 90 deg pitch angle 10-eV protons, also symmetric about the geomagnetic equator, were observed in conjunction with the waves. These may serve as a source of free energy for the pulsations. These observations lend support to recent studies suggesting that many dayside compressional wave events are related to localized field line resonance near plasmapauselike boundaries, but also include features that cannot be explained by existing theories.

Inductive-dynamic magnetosphere-ionosphere coupling via MHD waves

NASA Astrophysics Data System (ADS)

Tu, Jiannan; Song, Paul; Vasyliūnas, Vytenis M.

2014-01-01

In the present study, we investigate magnetosphere-ionosphere/thermosphere (M-IT) coupling via MHD waves by numerically solving time-dependent continuity, momentum, and energy equations for ions and neutrals, together with Maxwell's equations (Ampère's and Faraday's laws) and with photochemistry included. This inductive-dynamic approach we use is fundamentally different from those in previous magnetosphere-ionosphere (M-I) coupling models: all MHD wave modes are retained, and energy and momentum exchange between waves and plasma are incorporated into the governing equations, allowing a self-consistent examination of dynamic M-I coupling. Simulations, using an implicit numerical scheme, of the 1-D ionosphere/thermosphere system responding to an imposed convection velocity at the top boundary are presented to show how magnetosphere and ionosphere are coupled through Alfvén waves during the transient stage when the IT system changes from one quasi steady state to another. Wave reflection from the low-altitude ionosphere plays an essential role, causing overshoots and oscillations of ionospheric perturbations, and the dynamical Hall effect is an inherent aspect of the M-I coupling. The simulations demonstrate that the ionosphere/thermosphere responds to magnetospheric driving forces as a damped oscillator.

Time-Harmonic Gaussian Beams: Exact Solutions of the Helmhotz Equation in Free Space

NASA Astrophysics Data System (ADS)

Kiselev, A. P.

2017-12-01

An exact solution of the Helmholtz equation u xx + u yy + u zz + k 2 u = 0 is presented, which describes propagation of monochromatic waves in the free space. The solution has the form of a superposition of plane waves with a specific weight function dependent on a certain free parameter a. If ka→∞, the solution is localized in the Gaussian manner in a vicinity of a certain straight line and asymptotically coincides with the famous approximate solution known as the fundamental mode of a paraxial Gaussian beam. The asymptotics of the aforementioned exact solution does not include a backward wave.

Kinetic effects on Alfven wave nonlinearity. II - The modified nonlinear wave equation

NASA Technical Reports Server (NTRS)

Spangler, Steven R.

1990-01-01

A previously developed Vlasov theory is used here to study the role of resonant particle and other kinetic effects on Alfven wave nonlinearity. A hybrid fluid-Vlasov equation approach is used to obtain a modified version of the derivative nonlinear Schroedinger equation. The differences between a scalar model for the plasma pressure and a tensor model are discussed. The susceptibilty of the modified nonlinear wave equation to modulational instability is studied. The modulational instability normally associated with the derivative nonlinear Schroedinger equation will, under most circumstances, be restricted to left circularly polarized waves. The nonlocal term in the modified nonlinear wave equation engenders a new modulational instability that is independent of beta and the sense of circular polarization. This new instability may explain the occurrence of wave packet steepening for all values of the plasma beta in the vicinity of the earth's bow shock.

Group theoretical derivation of the minimal coupling principle

NASA Astrophysics Data System (ADS)

Nisticò, Giuseppe

2017-04-01

The group theoretical methods worked out by Bargmann, Mackey and Wigner, which deductively establish the Quantum Theory of a free particle for which Galileian transformations form a symmetry group, are extended to the case of an interacting particle. In doing so, the obstacles caused by loss of symmetry are overcome. In this approach, specific forms of the wave equation of an interacting particle, including the equation derived from the minimal coupling principle, are implied by particular first-order invariance properties that characterize the interaction with respect to specific subgroups of Galileian transformations; moreover, the possibility of yet unknown forms of the wave equation is left open.

Traveling-Wave Solutions of the Kolmogorov-Petrovskii-Piskunov Equation

NASA Astrophysics Data System (ADS)

Pikulin, S. V.

2018-02-01

We consider quasi-stationary solutions of a problem without initial conditions for the Kolmogorov-Petrovskii-Piskunov (KPP) equation, which is a quasilinear parabolic one arising in the modeling of certain reaction-diffusion processes in the theory of combustion, mathematical biology, and other areas of natural sciences. A new efficiently numerically implementable analytical representation is constructed for self-similar plane traveling-wave solutions of the KPP equation with a special right-hand side. Sufficient conditions for an auxiliary function involved in this representation to be analytical for all values of its argument, including the endpoints, are obtained. Numerical results are obtained for model examples.

Time-Reversal Generation of Rogue Waves

NASA Astrophysics Data System (ADS)

Chabchoub, Amin; Fink, Mathias

2014-03-01

The formation of extreme localizations in nonlinear dispersive media can be explained and described within the framework of nonlinear evolution equations, such as the nonlinear Schrödinger equation (NLS). Within the class of exact NLS breather solutions on a finite background, which describe the modulational instability of monochromatic wave trains, the hierarchy of rational solutions localized in both time and space is considered to provide appropriate prototypes to model rogue wave dynamics. Here, we use the time-reversal invariance of the NLS to propose and experimentally demonstrate a new approach to constructing strongly nonlinear localized waves focused in both time and space. The potential applications of this time-reversal approach include remote sensing and motivated analogous experimental analysis in other nonlinear dispersive media, such as optics, Bose-Einstein condensates, and plasma, where the wave motion dynamics is governed by the NLS.

Influence of nonlinear interactions on the development of instability in hydrodynamic wave systems

DOE Office of Scientific and Technical Information (OSTI.GOV)

Romanova, N. N.; Chkhetiani, O. G., E-mail: ochkheti@mx.iki.rssi.ru, E-mail: ochkheti@gmail.ru; Yakushkin, I. G.

2016-05-15

The problem of the development of shear instability in a three-layer medium simulating the flow of a stratified incompressible fluid is considered. The hydrodynamic equations are solved by expanding the Hamiltonian in a small parameter. The equations for three interacting waves, one of which is unstable, have been derived and solved numerically. The three-wave interaction is shown to stabilize the instability. Various regimes of the system’s dynamics, including the stochastic ones dependent on one of the invariants in the problem, can arise in this case. It is pointed out that the instability development scenario considered differs from the previously consideredmore » scenario of a different type, where the three-wave interaction does not stabilize the instability. The interaction of wave packets is considered briefly.« less

Calculation of the Full Scattering Amplitude without Partial Wave Decomposition II

NASA Technical Reports Server (NTRS)

Shertzer, J.; Temkin, A.

2003-01-01

As is well known, the full scattering amplitude can be expressed as an integral involving the complete scattering wave function. We have shown that the integral can be simplified and used in a practical way. Initial application to electron-hydrogen scattering without exchange was highly successful. The Schrodinger equation (SE) can be reduced to a 2d partial differential equation (pde), and was solved using the finite element method. We have now included exchange by solving the resultant SE, in the static exchange approximation. The resultant equation can be reduced to a pair of coupled pde's, to which the finite element method can still be applied. The resultant scattering amplitudes, both singlet and triplet, as a function of angle can be calculated for various energies. The results are in excellent agreement with converged partial wave results.

The Effect of Orifice Eccentricity on Instability of Liquid Jets

NASA Astrophysics Data System (ADS)

Amini, Ghobad; Dolatabadi, Ali

2011-11-01

The hydrodynamic instability of inviscid jets issuing from elliptic orifices is studied. A linear stability analysis is presented for liquid jets that includes the effect of the surrounding gas and an explicit dispersion equation is derived for waves on an infinite uniform jet column. Elliptic configuration has two extreme cases; round jet when ratio of minor to major axis is unity and plane sheet when this ratio approaches zero. Dispersion equation of elliptic jet is approximated for large and small aspect ratios considering asymptotic of the dispersion equation. In case of aspect ratio equal to one, the dispersion equation is analogous to one of the circular jets derived by Yang. In case of aspect ratio approaches zero, the behavior of waves is qualitatively similar to that of long waves on a two dimensional liquid jets and the varicose and sinuous modes are predicted. The growth rate of initial disturbances for various azimuthal modes has been presented in a wide range of disturbances. PhD Candidate.

Extended nonlinear Schrödinger equation with higher-order odd and even terms and its rogue wave solutions.

PubMed

Ankiewicz, Adrian; Wang, Yan; Wabnitz, Stefan; Akhmediev, Nail

2014-01-01

We consider an extended nonlinear Schrödinger equation with higher-order odd (third order) and even (fourth order) terms with variable coefficients. The resulting equation has soliton solutions and approximate rogue wave solutions. We present these solutions up to second order. Moreover, specific constraints on the parameters of higher-order terms provide integrability of the resulting equation, providing a corresponding Lax pair. Particular cases of this equation are the Hirota and the Lakshmanan-Porsezian-Daniel equations. The resulting integrable equation admits exact rogue wave solutions. In particular cases, mentioned above, these solutions are reduced to the rogue wave solutions of the corresponding equations.

A mixture-energy-consistent six-equation two-phase numerical model for fluids with interfaces, cavitation and evaporation waves

DOE Office of Scientific and Technical Information (OSTI.GOV)

Pelanti, Marica, E-mail: marica.pelanti@ensta-paristech.fr; Shyue, Keh-Ming, E-mail: shyue@ntu.edu.tw

2014-02-15

We model liquid–gas flows with cavitation by a variant of the six-equation single-velocity two-phase model with stiff mechanical relaxation of Saurel–Petitpas–Berry (Saurel et al., 2009) [9]. In our approach we employ phasic total energy equations instead of the phasic internal energy equations of the classical six-equation system. This alternative formulation allows us to easily design a simple numerical method that ensures consistency with mixture total energy conservation at the discrete level and agreement of the relaxed pressure at equilibrium with the correct mixture equation of state. Temperature and Gibbs free energy exchange terms are included in the equations as relaxationmore » terms to model heat and mass transfer and hence liquid–vapor transition. The algorithm uses a high-resolution wave propagation method for the numerical approximation of the homogeneous hyperbolic portion of the model. In two dimensions a fully-discretized scheme based on a hybrid HLLC/Roe Riemann solver is employed. Thermo-chemical terms are handled numerically via a stiff relaxation solver that forces thermodynamic equilibrium at liquid–vapor interfaces under metastable conditions. We present numerical results of sample tests in one and two space dimensions that show the ability of the proposed model to describe cavitation mechanisms and evaporation wave dynamics.« less

Several reverse-time integrable nonlocal nonlinear equations: Rogue-wave solutions

NASA Astrophysics Data System (ADS)

Yang, Bo; Chen, Yong

2018-05-01

A study of rogue-wave solutions in the reverse-time nonlocal nonlinear Schrödinger (NLS) and nonlocal Davey-Stewartson (DS) equations is presented. By using Darboux transformation (DT) method, several types of rogue-wave solutions are constructed. Dynamics of these rogue-wave solutions are further explored. It is shown that the (1 + 1)-dimensional fundamental rogue-wave solutions in the reverse-time NLS equation can be globally bounded or have finite-time blowing-ups. It is also shown that the (2 + 1)-dimensional line rogue waves in the reverse-time nonlocal DS equations can be bounded for all space and time or develop singularities in critical time. In addition, the multi- and higher-order rogue waves exhibit richer structures, most of which have no counterparts in the corresponding local nonlinear equations.

Multiple branches of travelling waves for the Gross–Pitaevskii equation

NASA Astrophysics Data System (ADS)

Chiron, David; Scheid, Claire

2018-06-01

Explicit solitary waves are known to exist for the Kadomtsev–Petviashvili-I (KP-I) equation in dimension 2. We first address numerically the question of their Morse index. The results confirm that the lump solitary wave has Morse index one and that the other explicit solutions correspond to excited states. We then turn to the 2D Gross–Pitaevskii (GP) equation, which in some long wave regime converges to the KP-I equation. Numerical simulations have already shown that a branch of travelling waves of GP converges to a ground state of KP-I, expected to be the lump. In this work, we perform numerical simulations showing that other explicit solitary waves solutions to the KP-I equation give rise to new branches of travelling waves of GP corresponding to excited states.

Computational methods and traveling wave solutions for the fourth-order nonlinear Ablowitz-Kaup-Newell-Segur water wave dynamical equation via two methods and its applications

NASA Astrophysics Data System (ADS)

Ali, Asghar; Seadawy, Aly R.; Lu, Dianchen

2018-05-01

The aim of this article is to construct some new traveling wave solutions and investigate localized structures for fourth-order nonlinear Ablowitz-Kaup-Newell-Segur (AKNS) water wave dynamical equation. The simple equation method (SEM) and the modified simple equation method (MSEM) are applied in this paper to construct the analytical traveling wave solutions of AKNS equation. The different waves solutions are derived by assigning special values to the parameters. The obtained results have their importance in the field of physics and other areas of applied sciences. All the solutions are also graphically represented. The constructed results are often helpful for studying several new localized structures and the waves interaction in the high-dimensional models.

Shock Waves in a Bose-Einstein Condensate

NASA Technical Reports Server (NTRS)

Kulikov, Igor; Zak, Michail

2005-01-01

A paper presents a theoretical study of shock waves in a trapped Bose-Einstein condensate (BEC). The mathematical model of the BEC in this study is a nonlinear Schroedinger equation (NLSE) in which (1) the role of the wave function of a single particle in the traditional Schroedinger equation is played by a space- and time-dependent complex order parameter (x,t) proportional to the square root of the density of atoms and (2) the atoms engage in a repulsive interaction characterized by a potential proportional to | (x,t)|2. Equations that describe macroscopic perturbations of the BEC at zero temperature are derived from the NLSE and simplifying assumptions are made, leading to equations for the propagation of sound waves and the transformation of sound waves into shock waves. Equations for the speeds of shock waves and the relationships between jumps of velocity and density across shock fronts are derived. Similarities and differences between this theory and the classical theory of sound waves and shocks in ordinary gases are noted. The present theory is illustrated by solving the equations for the example of a shock wave propagating in a cigar-shaped BEC.

Internal Solitons in the Oceans

DTIC Science & Technology

2006-01-01

stratification and also allow various generalizations of the KdV equa- tion, such as the Kadomtsev - Petviashvili equation shown below. The soliton... Kadomtsev - Petviashvili (KP) equation , which is applicable to a weakly diffracted wave beam, and is based again on adding a small term to the KdV equation ...well-known Boussinesq and Korteweg-de Vries equations . Then certain generalizations are considered, including effects of cubic nonlin- earity, Earth’s

Numerical study of the Kadomtsev-Petviashvili equation and dispersive shock waves

NASA Astrophysics Data System (ADS)

Grava, T.; Klein, C.; Pitton, G.

2018-02-01

A detailed numerical study of the long time behaviour of dispersive shock waves in solutions to the Kadomtsev-Petviashvili (KP) I equation is presented. It is shown that modulated lump solutions emerge from the dispersive shock waves. For the description of dispersive shock waves, Whitham modulation equations for KP are obtained. It is shown that the modulation equations near the soliton line are hyperbolic for the KPII equation while they are elliptic for the KPI equation leading to a focusing effect and the formation of lumps. Such a behaviour is similar to the appearance of breathers for the focusing nonlinear Schrödinger equation in the semiclassical limit.

Alfvén Wave Reflection and Turbulent Heating in the Solar Wind from 1 Solar Radius to 1 AU: An Analytical Treatment

NASA Astrophysics Data System (ADS)

Chandran, Benjamin D. G.; Hollweg, Joseph V.

2009-12-01

We study the propagation, reflection, and turbulent dissipation of Alfvén waves in coronal holes and the solar wind. We start with the Heinemann-Olbert equations, which describe non-compressive magnetohydrodynamic fluctuations in an inhomogeneous medium with a background flow parallel to the background magnetic field. Following the approach of Dmitruk et al., we model the nonlinear terms in these equations using a simple phenomenology for the cascade and dissipation of wave energy and assume that there is much more energy in waves propagating away from the Sun than waves propagating toward the Sun. We then solve the equations analytically for waves with periods of hours and longer to obtain expressions for the wave amplitudes and turbulent heating rate as a function of heliocentric distance. We also develop a second approximate model that includes waves with periods of roughly one minute to one hour, which undergo less reflection than the longer-period waves, and compare our models to observations. Our models generalize the phenomenological model of Dmitruk et al. by accounting for the solar wind velocity, so that the turbulent heating rate can be evaluated from the coronal base out past the Alfvén critical point—that is, throughout the region in which most of the heating and acceleration occurs. The simple analytical expressions that we obtain can be used to incorporate Alfvén-wave reflection and turbulent heating into fluid models of the solar wind.

Applications of exact traveling wave solutions of Modified Liouville and the Symmetric Regularized Long Wave equations via two new techniques

NASA Astrophysics Data System (ADS)

Lu, Dianchen; Seadawy, Aly R.; Ali, Asghar

2018-06-01

In this current work, we employ novel methods to find the exact travelling wave solutions of Modified Liouville equation and the Symmetric Regularized Long Wave equation, which are called extended simple equation and exp(-Ψ(ξ))-expansion methods. By assigning the different values to the parameters, different types of the solitary wave solutions are derived from the exact traveling wave solutions, which shows the efficiency and precision of our methods. Some solutions have been represented by graphical. The obtained results have several applications in physical science.

Nonlocal symmetries, solitary waves and cnoidal periodic waves of the (2+1)-dimensional breaking soliton equation

NASA Astrophysics Data System (ADS)

Zou, Li; Tian, Shou-Fu; Feng, Lian-Li

2017-12-01

In this paper, we consider the (2+1)-dimensional breaking soliton equation, which describes the interaction of a Riemann wave propagating along the y-axis with a long wave along the x-axis. By virtue of the truncated Painlevé expansion method, we obtain the nonlocal symmetry, Bäcklund transformation and Schwarzian form of the equation. Furthermore, by using the consistent Riccati expansion (CRE), we prove that the breaking soliton equation is solvable. Based on the consistent tan-function expansion, we explicitly derive the interaction solutions between solitary waves and cnoidal periodic waves.

Mode Identification of High-Amplitude Pressure Waves in Liquid Rocket Engines

NASA Astrophysics Data System (ADS)

EBRAHIMI, R.; MAZAHERI, K.; GHAFOURIAN, A.

2000-01-01

Identification of existing instability modes from experimental pressure measurements of rocket engines is difficult, specially when steep waves are present. Actual pressure waves are often non-linear and include steep shocks followed by gradual expansions. It is generally believed that interaction of these non-linear waves is difficult to analyze. A method of mode identification is introduced. After presumption of constituent modes, they are superposed by using a standard finite difference scheme for solution of the classical wave equation. Waves are numerically produced at each end of the combustion tube with different wavelengths, amplitudes, and phases with respect to each other. Pressure amplitude histories and phase diagrams along the tube are computed. To determine the validity of the presented method for steep non-linear waves, the Euler equations are numerically solved for non-linear waves, and negligible interactions between these waves are observed. To show the applicability of this method, other's experimental results in which modes were identified are used. Results indicate that this simple method can be used in analyzing complicated pressure signal measurements.

Parameterization of planetary wave breaking in the middle atmosphere

NASA Technical Reports Server (NTRS)

Garcia, Rolando R.

1991-01-01

A parameterization of planetary wave breaking in the middle atmosphere has been developed and tested in a numerical model which includes governing equations for a single wave and the zonal-mean state. The parameterization is based on the assumption that wave breaking represents a steady-state equilibrium between the flux of wave activity and its dissipation by nonlinear processes, and that the latter can be represented as linear damping of the primary wave. With this and the additional assumption that the effect of breaking is to prevent further amplitude growth, the required dissipation rate is readily obtained from the steady-state equation for wave activity; diffusivity coefficients then follow from the dissipation rate. The assumptions made in the derivation are equivalent to those commonly used in parameterizations for gravity wave breaking, but the formulation in terms of wave activity helps highlight the central role of the wave group velocity in determining the dissipation rate. Comparison of model results with nonlinear calculations of wave breaking and with diagnostic determinations of stratospheric diffusion coefficients reveals remarkably good agreement, and suggests that the parameterization could be useful for simulating inexpensively, but realistically, the effects of planetary wave transport.

Modeling of Wave Spectrum and Wave Breaking Statistics Based on Balance Equation

NASA Astrophysics Data System (ADS)

Irisov, V.

2012-12-01

Surface roughness and foam coverage are the parameters determining microwave emissivity of sea surface in a wide range of wind. Existing empirical wave spectra are not associated with wave breaking statistics although physically they are closely related. We propose a model of sea surface based on the balance of three terms: wind input, dissipation, and nonlinear wave-wave interaction. It provides an insight on wave generation, interaction, and dissipation - very important parameters for understanding of wave development under changing oceanic and atmospheric conditions. The wind input term is the best known among all three. For our analysis we assume a wind input term as it was proposed by Plant [1982] and consider modification necessary to do to account for proper interaction of long fast waves with wind. For long gravity waves (longer than 15-30 cm) the dissipation term can be related to the wave breaking with whitecaps, as it was shown by Kudryavtsev et al. [2003], so we assume the cubic dependence of dissipation term on wind. It implies certain limitations on the spectrum shape. The most difficult is to estimate the term describing nonlinear wave-wave interaction. Hasselmann [1962] and Zakharov [1999] developed theory of 4-wave interaction, but the resulting equation requires at least 3-fold integration over wavenumbers at each time step of integration of balance equation, which makes it difficult for direct numerical modeling. It is desirable to use an approximation of wave-wave interaction term, which preserves wave action, energy, and momentum, and can be easily estimated during time integration of balance equation. Zakharov and Pushkarev [1999] proposed the diffusion approximation of the wave interaction term and showed that it can be used for estimate of wave spectrum. We believe their assumption that wave-wave interaction is the dominant factor in forming the wave spectrum does not agree with the observations made by Hwang and Sletten [2008]. Finally we consider modifications of the model equation, which can be done to describe gravity-capillary and capillary waves. An obvious correction is to add viscous dissipation. A little less obvious is a transition from 4-wave to 3-wave interaction. The model allows one to include easily generation of parasitic capillary waves as it was proposed by Kudryavtsev et al. [2003]. A modification of dissipation term can explain an "overshoot" phenomenon observed in JONSWAP spectrum. These examples demonstrate that the proposed model is quite flexible and can be used to account for various physical phenomena. The resulting balance equation is easy to integrate using a personal computer and necessity of its numerical solution is paid by the model flexibility and better physical background compared with empirical spectra. References Hasselmann, K., J. Fluid Mech., 12, pp.481-500, 1962. Hwang, P., and M. Sletten, J. Geophys. Res., 113, doi:10.1029/2007JC004277, 2008. Kudryavtsev, V., et al., J. Geophys. Res., 108 (C3), doi:10.1029/2001JC001003, 2003. Plant, W. J., J. Geophys. Res., vol. 87, pp. 1961-1967, 1982. Zakharov, V., and A. Pushkarev, Nonlinear Processes in Geophysics, 6, pp.1-10, 1999. Zakharov, V., Eur. J. Mech. B/Fluids, 18, pp.327-344, 1999.

Detecting Moving Targets by Use of Soliton Resonances

NASA Technical Reports Server (NTRS)

Zak, Michael; Kulikov, Igor

2003-01-01

A proposed method of detecting moving targets in scenes that include cluttered or noisy backgrounds is based on a soliton-resonance mathematical model. The model is derived from asymptotic solutions of the cubic Schroedinger equation for a one-dimensional system excited by a position-and-time-dependent externally applied potential. The cubic Schroedinger equation has general significance for time-dependent dispersive waves. It has been used to approximate several phenomena in classical as well as quantum physics, including modulated beams in nonlinear optics, and superfluids (in particular, Bose-Einstein condensates). In the proposed method, one would take advantage of resonant interactions between (1) a soliton excited by the position-and-time-dependent potential associated with a moving target and (2) eigen-solitons, which represent dispersive waves and are solutions of the cubic Schroedinger equation for a time-independent potential.

Rogue-wave solutions of the Zakharov equation

NASA Astrophysics Data System (ADS)

Rao, Jiguang; Wang, Lihong; Liu, Wei; He, Jingsong

2017-12-01

Using the bilinear transformation method, we derive general rogue-wave solutions of the Zakharov equation. We present these Nth-order rogue-wave solutions explicitly in terms of Nth-order determinants whose matrix elements have simple expressions. We show that the fundamental rogue wave is a line rogue wave with a line profile on the plane ( x, y) arising from a constant background at t ≪ 0 and then gradually tending to the constant background for t ≫ 0. Higher-order rogue waves arising from a constant background and later disappearing into it describe the interaction of several fundamental line rogue waves. We also consider different structures of higher-order rogue waves. We present differences between rogue waves of the Zakharov equation and of the first type of the Davey-Stewartson equation analytically and graphically.

Evidence for self-refraction in a convergence zone: NPE (Nonlinear progressive wave equation) model results

NASA Technical Reports Server (NTRS)

Mcdonald, B. Edward; Plante, Daniel R.

1989-01-01

The nonlinear progressive wave equation (NPE) model was developed by the Naval Ocean Research and Development Activity during 1982 to 1987 to study nonlinear effects in long range oceanic propagation of finite amplitude acoustic waves, including weak shocks. The NPE model was applied to propagation of a generic shock wave (initial condition provided by Sandia Division 1533) in a few illustrative environments. The following consequences of nonlinearity are seen by comparing linear and nonlinear NPE results: (1) a decrease in shock strength versus range (a well-known result of entropy increases at the shock front); (2) an increase in the convergence zone range; and (3) a vertical meandering of the energy path about the corresponding linear ray path. Items (2) and (3) are manifestations of self-refraction.

Pressure and tension waves from bubble collapse near a solid boundary: A numerical approach.

PubMed

Lechner, Christiane; Koch, Max; Lauterborn, Werner; Mettin, Robert

2017-12-01

The acoustic waves being generated during the motion of a bubble in water near a solid boundary are calculated numerically. The open source package OpenFOAM is used for solving the Navier-Stokes equation and extended to include nonlinear acoustic wave effects via the Tait equation for water. A bubble model with a small amount of gas is chosen, the gas obeying an adiabatic law. A bubble starting from a small size with high internal pressure near a flat, solid boundary is studied. The sequence of events from bubble growth via axial microjet formation, jet impact, annular nanojet formation, torus-bubble collapse, and bubble rebound to second collapse is described. The different pressure and tension waves with their propagation properties are demonstrated.

Characteristics of solitary waves, quasiperiodic solutions, homoclinic breather solutions and rogue waves in the generalized variable-coefficient forced Kadomtsev-Petviashvili equation

NASA Astrophysics Data System (ADS)

Yan, Xue-Wei; Tian, Shou-Fu; Dong, Min-Jie; Zou, Li

2017-12-01

In this paper, the generalized variable-coefficient forced Kadomtsev-Petviashvili (gvcfKP) equation is investigated, which can be used to characterize the water waves of long wavelength relating to nonlinear restoring forces. Using a dependent variable transformation and combining the Bell’s polynomials, we accurately derive the bilinear expression for the gvcfKP equation. By virtue of bilinear expression, its solitary waves are computed in a very direct method. By using the Riemann theta function, we derive the quasiperiodic solutions for the equation under some limitation factors. Besides, an effective way can be used to calculate its homoclinic breather waves and rogue waves, respectively, by using an extended homoclinic test function. We hope that our results can help enrich the dynamical behavior of the nonlinear wave equations with variable-coefficient.

Nonlinear modes of the tensor Dirac equation and CPT violation

NASA Technical Reports Server (NTRS)

Reifler, Frank J.; Morris, Randall D.

1993-01-01

Recently, it has been shown that Dirac's bispinor equation can be expressed, in an equivalent tensor form, as a constrained Yang-Mills equation in the limit of an infinitely large coupling constant. It was also shown that the free tensor Dirac equation is a completely integrable Hamiltonian system with Lie algebra type Poisson brackets, from which Fermi quantization can be derived directly without using bispinors. The Yang-Mills equation for a finite coupling constant is investigated. It is shown that the nonlinear Yang-Mills equation has exact plane wave solutions in one-to-one correspondence with the plane wave solutions of Dirac's bispinor equation. The theory of nonlinear dispersive waves is applied to establish the existence of wave packets. The CPT violation of these nonlinear wave packets, which could lead to new observable effects consistent with current experimental bounds, is investigated.

Classification of the Lie and Noether point symmetries for the Wave and the Klein-Gordon equations in pp-wave spacetimes

NASA Astrophysics Data System (ADS)

Paliathanasis, A.; Tsamparlis, M.; Mustafa, M. T.

2018-02-01

A complete classification of the Lie and Noether point symmetries for the Klein-Gordon and the wave equation in pp-wave spacetimes is obtained. The classification analysis is carried out by reducing the problem of the determination of the point symmetries to the problem of existence of conformal killing vectors on the pp-wave spacetimes. Employing the existing results for the isometry classes of the pp-wave spacetimes, the functional form of the potential is determined for which the Klein-Gordon equation admits point symmetries and Noetherian conservation law. Finally the Lie and Noether point symmetries of the wave equation are derived.

Some simple solutions of Schrödinger's equation for a free particle or for an oscillator

NASA Astrophysics Data System (ADS)

Andrews, Mark

2018-05-01

For a non-relativistic free particle, we show that the evolution of some simple initial wave functions made up of linear segments can be expressed in terms of Fresnel integrals. Examples include the square wave function and the triangular wave function. The method is then extended to wave functions made from quadratic elements. The evolution of all these initial wave functions can also be found for the harmonic oscillator by a transformation of the free evolutions.

Traveling wave solutions of the nonlinear Schrödinger equation

NASA Astrophysics Data System (ADS)

Akbari-Moghanjoughi, M.

2017-10-01

In this paper, we investigate the traveling soliton and the periodic wave solutions of the nonlinear Schrödinger equation (NLSE) with generalized nonlinear functionality. We also explore the underlying close connection between the well-known KdV equation and the NLSE. It is remarked that both one-dimensional KdV and NLSE models share the same pseudoenergy spectrum. We also derive the traveling wave solutions for two cases of weakly nonlinear mathematical models, namely, the Helmholtz and the Duffing oscillators' potentials. It is found that these models only allow gray-type NLSE solitary propagations. It is also found that the pseudofrequency ratio for the Helmholtz potential between the nonlinear periodic carrier and the modulated sinusoidal waves is always in the range 0.5 ≤ Ω/ω ≤ 0.537285 regardless of the potential parameter values. The values of Ω/ω = {0.5, 0.537285} correspond to the cnoidal waves modulus of m = {0, 1} for soliton and sinusoidal limits and m = 0.5, respectively. Moreover, the current NLSE model is extended to fully NLSE (FNLSE) situation for Sagdeev oscillator pseudopotential which can be derived using a closed set of hydrodynamic fluid equations with a fully integrable Hamiltonian system. The generalized quasi-three-dimensional traveling wave solution is also derived. The current simple hydrodynamic plasma model may also be generalized to two dimensions and other complex situations including different charged species and cases with magnetic or gravitational field effects.

Alternative stable qP wave equations in TTI media with their applications for reverse time migration

NASA Astrophysics Data System (ADS)

Zhou, Yang; Wang, Huazhong; Liu, Wenqing

2015-10-01

Numerical instabilities may arise if the spatial variation of symmetry axis is handled improperly when implementing P-wave modeling and reverse time migration in heterogeneous tilted transversely isotropic (TTI) media, especially in the cases where fast changes exist in TTI symmetry axis’ directions. Based on the pseudo-acoustic approximation to anisotropic elastic wave equations in Cartesian coordinates, alternative second order qP (quasi-P) wave equations in TTI media are derived in this paper. Compared with conventional stable qP wave equations, the proposed equations written in stress components contain only spatial derivatives of wavefield variables (stress components) and are free from spatial derivatives involving media parameters. These lead to an easy and efficient implementation for stable P-wave modeling and imaging. Numerical experiments demonstrate the stability and computational efficiency of the presented equations in complex TTI media.

The effect of latent heat release on synoptic-to-planetary wave interactions and its implication for satellite observations: Theoretical modeling

NASA Technical Reports Server (NTRS)

Branscome, Lee E.; Bleck, Rainer; Obrien, Enda

1990-01-01

The project objectives are to develop process models to investigate the interaction of planetary and synoptic-scale waves including the effects of latent heat release (precipitation), nonlinear dynamics, physical and boundary-layer processes, and large-scale topography; to determine the importance of latent heat release for temporal variability and time-mean behavior of planetary and synoptic-scale waves; to compare the model results with available observations of planetary and synoptic wave variability; and to assess the implications of the results for monitoring precipitation in oceanic-storm tracks by satellite observing systems. Researchers have utilized two different models for this project: a two-level quasi-geostrophic model to study intraseasonal variability, anomalous circulations and the seasonal cycle, and a 10-level, multi-wave primitive equation model to validate the two-level Q-G model and examine effects of convection, surface processes, and spherical geometry. It explicitly resolves several planetary and synoptic waves and includes specific humidity (as a predicted variable), moist convection, and large-scale precipitation. In the past year researchers have concentrated on experiments with the multi-level primitive equation model. The dynamical part of that model is similar to the spectral model used by the National Meteorological Center for medium-range forecasts. The model includes parameterizations of large-scale condensation and moist convection. To test the validity of results regarding the influence of convective precipitation, researchers can use either one of two different convective schemes in the model, a Kuo convective scheme or a modified Arakawa-Schubert scheme which includes downdrafts. By choosing one or the other scheme, they can evaluate the impact of the convective parameterization on the circulation. In the past year researchers performed a variety of initial-value experiments with the primitive-equation model. Using initial conditions typical of climatological winter conditions, they examined the behavior of synoptic and planetary waves growing in moist and dry environments. Surface conditions were representative of a zonally averaged ocean. They found that moist convection associated with baroclinic wave development was confined to the subtropics.

Matter rogue waves in an F=1 spinor Bose-Einstein condensate.

PubMed

Qin, Zhenyun; Mu, Gui

2012-09-01

We report new types of matter rogue waves of a spinor (three-component) model of the Bose-Einstein condensate governed by a system of three nonlinearly coupled Gross-Pitaevskii equations. The exact first-order rational solutions containing one free parameter are obtained by means of a Darboux transformation for the integrable system where the mean-field interaction is attractive and the spin-exchange interaction is ferromagnetic. For different choices of the parameter, there exists a variety of different shaped solutions including two peaks in bright rogue waves and four dips in dark rogue waves. Furthermore, by utilizing the relation between the three-component and the one-component versions of the nonlinear Schrödinger equation, we can devise higher-order rational solutions, in which three components have different shapes. In addition, it is noteworthy that dark rogue wave features disappear in the third-order rational solution.

Modulation instability and dissipative rogue waves in ion-beam plasma: Roles of ionization, recombination, and electron attachment

DOE Office of Scientific and Technical Information (OSTI.GOV)

Guo, Shimin, E-mail: gsm861@126.com; Mei, Liquan, E-mail: lqmei@mail.xjtu.edu.cn

The amplitude modulation of ion-acoustic waves is investigated in an unmagnetized plasma containing positive ions, negative ions, and electrons obeying a kappa-type distribution that is penetrated by a positive ion beam. By considering dissipative mechanisms, including ionization, negative-positive ion recombination, and electron attachment, we introduce a comprehensive model for the plasma with the effects of sources and sinks. Via reductive perturbation theory, the modified nonlinear Schrödinger equation with a dissipative term is derived to govern the dynamics of the modulated waves. The effect of the plasma parameters on the modulation instability criterion for the modified nonlinear Schrödinger equation is numericallymore » investigated in detail. Within the unstable region, first- and second-order dissipative ion-acoustic rogue waves are present. The effect of the plasma parameters on the characteristics of the dissipative rogue waves is also discussed.« less

Simulations and analysis of asteroid-generated tsunamis using the shallow water equations

NASA Astrophysics Data System (ADS)

Berger, M. J.; LeVeque, R. J.; Weiss, R.

2016-12-01

We discuss tsunami propagation for asteroid-generated air bursts and water impacts. We present simulations for a range of conditions using the GeoClaw simulation software. Examples include meteors that span 5 to 250 MT of kinetic energy, and use bathymetry from the U.S. coastline. We also study radially symmetric one-dimensional equations to better explore the nature and decay rate of waves generated by air burst pressure disturbances traveling at the speed of sound in air, which is much greater than the gravity wave speed of the tsunami generated. One-dimensional simulations along a transect of bathymetry are also used to explore the resolution needed for the full two-dimensional simulations, which are much more expensive even with the use of adaptive mesh refinement due to the short wave lengths of these tsunamis. For this same reason, shallow water equations may be inadequate and we also discuss dispersive effects.

Self-dual form of Ruijsenaars-Schneider models and ILW equation with discrete Laplacian

NASA Astrophysics Data System (ADS)

Zabrodin, A.; Zotov, A.

2018-02-01

We discuss a self-dual form or the Bäcklund transformations for the continuous (in time variable) glN Ruijsenaars-Schneider model. It is based on the first order equations in N + M complex variables which include N positions of particles and M dual variables. The latter satisfy equations of motion of the glM Ruijsenaars-Schneider model. In the elliptic case it holds M = N while for the rational and trigonometric models M is not necessarily equal to N. Our consideration is similar to the previously obtained results for the Calogero-Moser models which are recovered in the non-relativistic limit. We also show that the self-dual description of the Ruijsenaars-Schneider models can be derived from complexified intermediate long wave equation with discrete Laplacian by means of the simple pole ansatz likewise the Calogero-Moser models arise from ordinary intermediate long wave and Benjamin-Ono equations.

Orbital stability of solitary waves for Kundu equation

NASA Astrophysics Data System (ADS)

Zhang, Weiguo; Qin, Yinghao; Zhao, Yan; Guo, Boling

In this paper, we consider the Kundu equation which is not a standard Hamiltonian system. The abstract orbital stability theory proposed by Grillakis et al. (1987, 1990) cannot be applied directly to study orbital stability of solitary waves for this equation. Motivated by the idea of Guo and Wu (1995), we construct three invariants of motion and use detailed spectral analysis to obtain orbital stability of solitary waves for Kundu equation. Since Kundu equation is more complex than the derivative Schrödinger equation, we utilize some techniques to overcome some difficulties in this paper. It should be pointed out that the results obtained in this paper are more general than those obtained by Guo and Wu (1995). We present a sufficient condition under which solitary waves are orbitally stable for 2c+sυ<0, while Guo and Wu (1995) only considered the case 2c+sυ>0. We obtain the results on orbital stability of solitary waves for the derivative Schrödinger equation given by Colin and Ohta (2006) as a corollary in this paper. Furthermore, we obtain orbital stability of solitary waves for Chen-Lee-Lin equation and Gerdjikov-Ivanov equation, respectively.

Consistent nonlinear deterministic and stochastic evolution equations for deep to shallow water wave shoaling

NASA Astrophysics Data System (ADS)

Vrecica, Teodor; Toledo, Yaron

2015-04-01

One-dimensional deterministic and stochastic evolution equations are derived for the dispersive nonlinear waves while taking dissipation of energy into account. The deterministic nonlinear evolution equations are formulated using operational calculus by following the approach of Bredmose et al. (2005). Their formulation is extended to include the linear and nonlinear effects of wave dissipation due to friction and breaking. The resulting equation set describes the linear evolution of the velocity potential for each wave harmonic coupled by quadratic nonlinear terms. These terms describe the nonlinear interactions between triads of waves, which represent the leading-order nonlinear effects in the near-shore region. The equations are translated to the amplitudes of the surface elevation by using the approach of Agnon and Sheremet (1997) with the correction of Eldeberky and Madsen (1999). The only current possibility for calculating the surface gravity wave field over large domains is by using stochastic wave evolution models. Hence, the above deterministic model is formulated as a stochastic one using the method of Agnon and Sheremet (1997) with two types of stochastic closure relations (Benney and Saffman's, 1966, and Hollway's, 1980). These formulations cannot be applied to the common wave forecasting models without further manipulation, as they include a non-local wave shoaling coefficients (i.e., ones that require integration along the wave rays). Therefore, a localization method was applied (see Stiassnie and Drimer, 2006, and Toledo and Agnon, 2012). This process essentially extracts the local terms that constitute the mean nonlinear energy transfer while discarding the remaining oscillatory terms, which transfer energy back and forth. One of the main findings of this work is the understanding that the approximated non-local coefficients behave in two essentially different manners. In intermediate water depths these coefficients indeed consist of rapidly oscillating terms, but as the water depth becomes shallow they change to an exponential growth (or decay) behavior. Hence, the formerly used localization technique cannot be justified for the shallow water region. A new formulation is devised for the localization in shallow water, it approximates the nonlinear non-local shoaling coefficient in shallow water and matches it to the one fitting to the intermediate water region. This allows the model behavior to be consistent from deep water to intermediate depths and up to the shallow water regime. Various simulations of the model were performed for the cases of intermediate, and shallow water, overall the model was found to give good results in both shallow and intermediate water depths. The essential difference between the shallow and intermediate nonlinear shoaling physics is explained via the dominating class III Bragg resonances phenomenon. By inspecting the resonance conditions and the nature of the dispersion relation, it is shown that unlike in the intermediate water regime, in shallow water depths the formation of resonant interactions is possible without taking into account bottom components. References Agnon, Y. & Sheremet, A. 1997 Stochastic nonlinear shoaling of directional spectra. J. Fluid Mech. 345, 79-99. Benney, D. J. & Saffman, P. G. 1966 Nonlinear interactions of random waves. Proc. R. Soc. Lond. A 289, 301-321. Bredmose, H., Agnon, Y., Madsen, P.A. & Schaffer, H.A. 2005 Wave transformation models with exact second-order transfer. European J. of Mech. - B/Fluids 24 (6), 659-682. Eldeberky, Y. & Madsen, P. A. 1999 Deterministic and stochastic evolution equations for fully dispersive and weakly nonlinear waves. Coastal Engineering 38, 1-24. Kaihatu, J. M. & Kirby, J. T. 1995 Nonlinear transformation of waves in infinite water depth. Phys. Fluids 8, 175-188. Holloway, G. 1980 Oceanic internal waves are not weak waves. J. Phys. Oceanogr. 10, 906-914. Stiassnie, M. & Drimer, N. 2006 Prediction of long forcing waves for harbor agitation studies. J. of waterways, port, coastal and ocean engineering 132(3), 166-171. Toledo, Y. & Agnon, Y. 2012 Stochastic evolution equations with localized nonlinear shoaling coefficients. European J. of Mech. - B/Fluids 34, 13-18.

Calculation of hypersonic shock structure using flux-split algorithms

NASA Technical Reports Server (NTRS)

Eppard, W. M.; Grossman, B.

1991-01-01

There exists an altitude regime in the atmosphere that is within the continuum domain, but wherein the conventional Navier-Stokes equations cease to be accurate. The altitude limits for this so called continuum transition regime depend on vehicle size and speed. Within this regime the thickness of the bow shock wave is no longer negligible when compared to the shock stand-off distance and the peak radiation intensity occurs within the shock wave structure itself. For this reason it is no longer valid to treat the shock wave as a discontinuous jump and it becomes necessary to compute through the shock wave itself. To accurately calculate hypersonic flowfields, the governing equations must be capable of yielding realistic profiles of flow variables throughout the structure of a hypersonic shock wave. The conventional form of the Navier-Stokes equations is restricted to flows with only small departures from translational equilibrium; it is for this reason they do not provide the capability to accurately predict hypersonic shock structure. Calculations in the continuum transition regime, therefore, require the use of governing equations other than Navier-Stokes. Several alternatives to Navier-Stokes are discussed; first for the case of a monatomic gas and then for the case of a diatomic gas where rotational energy must be included. Results are presented for normal shock calculations with argon and nitrogen.

Series expansion solutions for the multi-term time and space fractional partial differential equations in two- and three-dimensions

NASA Astrophysics Data System (ADS)

Ye, H.; Liu, F.; Turner, I.; Anh, V.; Burrage, K.

2013-09-01

Fractional partial differential equations with more than one fractional derivative in time describe some important physical phenomena, such as the telegraph equation, the power law wave equation, or the Szabo wave equation. In this paper, we consider two- and three-dimensional multi-term time and space fractional partial differential equations. The multi-term time-fractional derivative is defined in the Caputo sense, whose order belongs to the interval (1,2],(2,3],(3,4] or (0, m], and the space-fractional derivative is referred to as the fractional Laplacian form. We derive series expansion solutions based on a spectral representation of the Laplacian operator on a bounded region. Some applications are given for the two- and three-dimensional telegraph equation, power law wave equation and Szabo wave equation.

Travelling-wave solutions of a weakly nonlinear two-dimensional higher-order Kadomtsev-Petviashvili dynamical equation for dispersive shallow-water waves

NASA Astrophysics Data System (ADS)

Seadawy, Aly R.

2017-01-01

The propagation of three-dimensional nonlinear irrotational flow of an inviscid and incompressible fluid of the long waves in dispersive shallow-water approximation is analyzed. The problem formulation of the long waves in dispersive shallow-water approximation lead to fifth-order Kadomtsev-Petviashvili (KP) dynamical equation by applying the reductive perturbation theory. By using an extended auxiliary equation method, the solitary travelling-wave solutions of the two-dimensional nonlinear fifth-order KP dynamical equation are derived. An analytical as well as a numerical solution of the two-dimensional nonlinear KP equation are obtained and analyzed with the effects of external pressure flow.

Ring Current-Electromagnetic Ion Cyclotron Waves Coupling

NASA Technical Reports Server (NTRS)

Khazanov, G. V.

2005-01-01

The effect of Electromagnetic Ion Cyclotron (EMIC) waves, generated by ion temperature anisotropy in Earth s ring current (RC), is the best known example of wave- particle interaction in the magnetosphere. Also, there is much controversy over the importance of EMIC waves on RC depletion. Under certain conditions, relativistic electrons, with energies 21 MeV, can be removed from the outer radiation belt (RB) by EMIC wave scattering during a magnetic storm. That is why the calculation of EMIC waves must be a very critical part of the space weather studies. The new RC model that we have developed and present for the first time has several new features that we have combine together in a one single model: (a) several lower frequency cold plasma wave modes are taken into account; (b) wave tracing of these wave has been incorporated in the energy EMIC wave equation; (c) no assumptions regarding wave shape spectra have been made; (d) no assumptions regarding the shape of particle distribution have been made to calculate the growth rate; (e) pitch-angle, energy, and mix diffusions are taken into account together for the first time; (f) the exact loss-cone RC analytical solution has been found and coupled with bounce-averaged numerical solution of kinetic equation; (g) the EMIC waves saturation due to their modulation instability and LHW generation are included as an additional factor that contributes to this process; and (h) the hot ions were included in the real part of dielectric permittivity tensor. We compare our theoretical results with the different EMIC waves models as well as RC experimental data.

Super-rogue waves in simulations based on weakly nonlinear and fully nonlinear hydrodynamic equations.

PubMed

Slunyaev, A; Pelinovsky, E; Sergeeva, A; Chabchoub, A; Hoffmann, N; Onorato, M; Akhmediev, N

2013-07-01

The rogue wave solutions (rational multibreathers) of the nonlinear Schrödinger equation (NLS) are tested in numerical simulations of weakly nonlinear and fully nonlinear hydrodynamic equations. Only the lowest order solutions from 1 to 5 are considered. A higher accuracy of wave propagation in space is reached using the modified NLS equation, also known as the Dysthe equation. This numerical modeling allowed us to directly compare simulations with recent results of laboratory measurements in Chabchoub et al. [Phys. Rev. E 86, 056601 (2012)]. In order to achieve even higher physical accuracy, we employed fully nonlinear simulations of potential Euler equations. These simulations provided us with basic characteristics of long time evolution of rational solutions of the NLS equation in the case of near-breaking conditions. The analytic NLS solutions are found to describe the actual wave dynamics of steep waves reasonably well.

Local energy decay for linear wave equations with variable coefficients

NASA Astrophysics Data System (ADS)

Ikehata, Ryo

2005-06-01

A uniform local energy decay result is derived to the linear wave equation with spatial variable coefficients. We deal with this equation in an exterior domain with a star-shaped complement. Our advantage is that we do not assume any compactness of the support on the initial data, and its proof is quite simple. This generalizes a previous famous result due to Morawetz [The decay of solutions of the exterior initial-boundary value problem for the wave equation, Comm. Pure Appl. Math. 14 (1961) 561-568]. In order to prove local energy decay, we mainly apply two types of ideas due to Ikehata-Matsuyama [L2-behaviour of solutions to the linear heat and wave equations in exterior domains, Sci. Math. Japon. 55 (2002) 33-42] and Todorova-Yordanov [Critical exponent for a nonlinear wave equation with damping, J. Differential Equations 174 (2001) 464-489].

Pure quasi-P wave equation and numerical solution in 3D TTI media

NASA Astrophysics Data System (ADS)

Zhang, Jian-Min; He, Bing-Shou; Tang, Huai-Gu

2017-03-01

Based on the pure quasi-P wave equation in transverse isotropic media with a vertical symmetry axis (VTI media), a quasi-P wave equation is obtained in transverse isotropic media with a tilted symmetry axis (TTI media). This is achieved using projection transformation, which rotates the direction vector in the coordinate system of observation toward the direction vector for the coordinate system in which the z-component is parallel to the symmetry axis of the TTI media. The equation has a simple form, is easily calculated, is not influenced by the pseudo-shear wave, and can be calculated reliably when δ is greater than ɛ. The finite difference method is used to solve the equation. In addition, a perfectly matched layer (PML) absorbing boundary condition is obtained for the equation. Theoretical analysis and numerical simulation results with forward modeling prove that the equation can accurately simulate a quasi-P wave in TTI medium.

Self-consistent-field perturbation theory for the Schröautdinger equation

NASA Astrophysics Data System (ADS)

Goodson, David Z.

1997-06-01

A method is developed for using large-order perturbation theory to solve the systems of coupled differential equations that result from the variational solution of the Schröautdinger equation with wave functions of product form. This is a noniterative, computationally efficient way to solve self-consistent-field (SCF) equations. Possible applications include electronic structure calculations using products of functions of collective coordinates that include electron correlation, vibrational SCF calculations for coupled anharmonic oscillators with selective coupling of normal modes, and ab initio calculations of molecular vibration spectra without the Born-Oppenheimer approximation.

Nonlinear dynamics of resonant electrons interacting with coherent Langmuir waves

NASA Astrophysics Data System (ADS)

Tobita, Miwa; Omura, Yoshiharu

2018-03-01

We study the nonlinear dynamics of resonant particles interacting with coherent waves in space plasmas. Magnetospheric plasma waves such as whistler-mode chorus, electromagnetic ion cyclotron waves, and hiss emissions contain coherent wave structures with various discrete frequencies. Although these waves are electromagnetic, their interaction with resonant particles can be approximated by equations of motion for a charged particle in a one-dimensional electrostatic wave. The equations are expressed in the form of nonlinear pendulum equations. We perform test particle simulations of electrons in an electrostatic model with Langmuir waves and a non-oscillatory electric field. We solve equations of motion and study the dynamics of particles with different values of inhomogeneity factor S defined as a ratio of the non-oscillatory electric field intensity to the wave amplitude. The simulation results demonstrate deceleration/acceleration, thermalization, and trapping of particles through resonance with a single wave, two waves, and multiple waves. For two-wave and multiple-wave cases, we describe the wave-particle interaction as either coherent or incoherent based on the probability of nonlinear trapping.

On an Acoustic Wave Equation Arising in Non-Equilibrium Gasdynamics. Classroom Notes

ERIC Educational Resources Information Center

Chandran, Pallath

2004-01-01

The sixth-order wave equation governing the propagation of one-dimensional acoustic waves in a viscous, heat conducting gaseous medium subject to relaxation effects has been considered. It has been reduced to a system of lower order equations corresponding to the finite speeds occurring in the equation, following a method due to Whitham. The lower…

High-frequency homogenization for travelling waves in periodic media.

PubMed

Harutyunyan, Davit; Milton, Graeme W; Craster, Richard V

2016-07-01

We consider high-frequency homogenization in periodic media for travelling waves of several different equations: the wave equation for scalar-valued waves such as acoustics; the wave equation for vector-valued waves such as electromagnetism and elasticity; and a system that encompasses the Schrödinger equation. This homogenization applies when the wavelength is of the order of the size of the medium periodicity cell. The travelling wave is assumed to be the sum of two waves: a modulated Bloch carrier wave having crystal wavevector [Formula: see text] and frequency ω 1 plus a modulated Bloch carrier wave having crystal wavevector [Formula: see text] and frequency ω 2 . We derive effective equations for the modulating functions, and then prove that there is no coupling in the effective equations between the two different waves both in the scalar and the system cases. To be precise, we prove that there is no coupling unless ω 1 = ω 2 and [Formula: see text] where Λ =(λ 1 λ 2 …λ d ) is the periodicity cell of the medium and for any two vectors [Formula: see text] the product a ⊙ b is defined to be the vector ( a 1 b 1 , a 2 b 2 ,…, a d b d ). This last condition forces the carrier waves to be equivalent Bloch waves meaning that the coupling constants in the system of effective equations vanish. We use two-scale analysis and some new weak-convergence type lemmas. The analysis is not at the same level of rigour as that of Allaire and co-workers who use two-scale convergence theory to treat the problem, but has the advantage of simplicity which will allow it to be easily extended to the case where there is degeneracy of the Bloch eigenvalue.

Modulational instability, beak-shaped rogue waves, multi-dark-dark solitons and dynamics in pair-transition-coupled nonlinear Schrödinger equations.

PubMed

Zhang, Guoqiang; Yan, Zhenya; Wen, Xiao-Yong

2017-07-01

The integrable coupled nonlinear Schrödinger equations with four-wave mixing are investigated. We first explore the conditions for modulational instability of continuous waves of this system. Secondly, based on the generalized N -fold Darboux transformation (DT), beak-shaped higher-order rogue waves (RWs) and beak-shaped higher-order rogue wave pairs are derived for the coupled model with attractive interaction in terms of simple determinants. Moreover, we derive the simple multi-dark-dark and kink-shaped multi-dark-dark solitons for the coupled model with repulsive interaction through the generalizing DT. We explore their dynamics and classifications by different kinds of spatial-temporal distribution structures including triangular, pentagonal, 'claw-like' and heptagonal patterns. Finally, we perform the numerical simulations to predict that some dark solitons and RWs are stable enough to develop within a short time. The results would enrich our understanding on nonlinear excitations in many coupled nonlinear wave systems with transition coupling effects.

A more fundamental approach to the derivation of nonlinear acoustic wave equations with fractional loss operators (L).

PubMed

Prieur, Fabrice; Vilenskiy, Gregory; Holm, Sverre

2012-10-01

A corrected derivation of nonlinear wave propagation equations with fractional loss operators is presented. The fundamental approach is based on fractional formulations of the stress-strain and heat flux definitions but uses the energy equation and thermodynamic identities to link density and pressure instead of an erroneous fractional form of the entropy equation as done in Prieur and Holm ["Nonlinear acoustic wave equations with fractional loss operators," J. Acoust. Soc. Am. 130(3), 1125-1132 (2011)]. The loss operator of the obtained nonlinear wave equations differs from the previous derivations as well as the dispersion equation, but when approximating for low frequencies the expressions for the frequency dependent attenuation and velocity dispersion remain unchanged.

Multi-Hamiltonian structure of equations of hydrodynamic type

NASA Astrophysics Data System (ADS)

Gümral, H.; Nutku, Y.

1990-11-01

The discussion of the Hamiltonian structure of two-component equations of hydrodynamic type is completed by presenting the Hamiltonian operators for Euler's equation governing the motion of plane sound waves of finite amplitude and another quasilinear second-order wave equation. There exists a doubly infinite family of conserved Hamiltonians for the equations of gas dynamics that degenerate into one, namely, the Benney sequence, for shallow-water waves. Infinite sequences of conserved quantities for these equations are also presented. In the case of multicomponent equations of hydrodynamic type, it is shown, that Kodama's generalization of the shallow-water equations admits bi-Hamiltonian structure.

Wind Wave Behavior in Fetch and Depth Limited Estuaries

NASA Astrophysics Data System (ADS)

Karimpour, Arash; Chen, Qin; Twilley, Robert R.

2017-01-01

Wetland dominated estuaries serve as one of the most productive natural ecosystems through their ecological, economic and cultural services, such as nursery grounds for fisheries, nutrient sequestration, and ecotourism. The ongoing deterioration of wetland ecosystems in many shallow estuaries raises concerns about the contributing erosive processes and their roles in restraining coastal restoration efforts. Given the combination of wetlands and shallow bays as landscape components that determine the function of estuaries, successful restoration strategies require knowledge of wind wave behavior in fetch and depth limited water as a critical design feature. We experimentally evaluate physics of wind wave growth in fetch and depth limited estuaries. We demonstrate that wave growth rate in shallow estuaries is a function of wind fetch to water depth ratio, which helps to develop a new set of parametric wave growth equations. We find that the final stage of wave growth in shallow estuaries can be presented by a product of water depth and wave number, whereby their product approaches 1.363 as either depth or wave energy increases. Suggested wave growth equations and their asymptotic constraints establish the magnitude of wave forces acting on wetland erosion that must be included in ecosystem restoration design.

Wind Wave Behavior in Fetch and Depth Limited Estuaries

PubMed Central

Karimpour, Arash; Chen, Qin; Twilley, Robert R.

2017-01-01

Wetland dominated estuaries serve as one of the most productive natural ecosystems through their ecological, economic and cultural services, such as nursery grounds for fisheries, nutrient sequestration, and ecotourism. The ongoing deterioration of wetland ecosystems in many shallow estuaries raises concerns about the contributing erosive processes and their roles in restraining coastal restoration efforts. Given the combination of wetlands and shallow bays as landscape components that determine the function of estuaries, successful restoration strategies require knowledge of wind wave behavior in fetch and depth limited water as a critical design feature. We experimentally evaluate physics of wind wave growth in fetch and depth limited estuaries. We demonstrate that wave growth rate in shallow estuaries is a function of wind fetch to water depth ratio, which helps to develop a new set of parametric wave growth equations. We find that the final stage of wave growth in shallow estuaries can be presented by a product of water depth and wave number, whereby their product approaches 1.363 as either depth or wave energy increases. Suggested wave growth equations and their asymptotic constraints establish the magnitude of wave forces acting on wetland erosion that must be included in ecosystem restoration design. PMID:28098236

Wind Wave Behavior in Fetch and Depth Limited Estuaries.

PubMed

Karimpour, Arash; Chen, Qin; Twilley, Robert R

2017-01-18

Wetland dominated estuaries serve as one of the most productive natural ecosystems through their ecological, economic and cultural services, such as nursery grounds for fisheries, nutrient sequestration, and ecotourism. The ongoing deterioration of wetland ecosystems in many shallow estuaries raises concerns about the contributing erosive processes and their roles in restraining coastal restoration efforts. Given the combination of wetlands and shallow bays as landscape components that determine the function of estuaries, successful restoration strategies require knowledge of wind wave behavior in fetch and depth limited water as a critical design feature. We experimentally evaluate physics of wind wave growth in fetch and depth limited estuaries. We demonstrate that wave growth rate in shallow estuaries is a function of wind fetch to water depth ratio, which helps to develop a new set of parametric wave growth equations. We find that the final stage of wave growth in shallow estuaries can be presented by a product of water depth and wave number, whereby their product approaches 1.363 as either depth or wave energy increases. Suggested wave growth equations and their asymptotic constraints establish the magnitude of wave forces acting on wetland erosion that must be included in ecosystem restoration design.

Vector rogue waves and baseband modulation instability in the defocusing regime.

PubMed

Baronio, Fabio; Conforti, Matteo; Degasperis, Antonio; Lombardo, Sara; Onorato, Miguel; Wabnitz, Stefan

2014-07-18

We report and discuss analytical solutions of the vector nonlinear Schrödinger equation that describe rogue waves in the defocusing regime. This family of solutions includes bright-dark and dark-dark rogue waves. The link between modulational instability (MI) and rogue waves is displayed by showing that only a peculiar kind of MI, namely baseband MI, can sustain rogue-wave formation. The existence of vector rogue waves in the defocusing regime is expected to be a crucial progress in explaining extreme waves in a variety of physical scenarios described by multicomponent systems, from oceanography to optics and plasma physics.

Nonparaxial wave beams and packets with general astigmatism

NASA Astrophysics Data System (ADS)

Kiselev, A. P.; Plachenov, A. B.; Chamorro-Posada, P.

2012-04-01

We present exact solutions of the wave equation involving an arbitrary wave form with a phase closely similar to the general astigmatic phase of paraxial wave optics. Special choices of the wave form allow general astigmatic beamlike and pulselike waves with a Gaussian-type unrestricted localization in space and time. These solutions are generalizations of the known Bateman-type waves obtained from the connection existing between beamlike solutions of the paraxial parabolic equation and relatively undistorted wave solutions of the wave equation. As a technical tool, we present a full description of parametrizations of 2×2 symmetric matrices with positive imaginary part, which arise in the theory of Gaussian beams.

Stability of dust ion acoustic solitary waves in a collisionless unmagnetized nonthermal plasma in presence of isothermal positrons

DOE Office of Scientific and Technical Information (OSTI.GOV)

Sardar, Sankirtan; Bandyopadhyay, Anup, E-mail: abandyopadhyay1965@gmail.com; Das, K. P.

A three-dimensional KP (Kadomtsev Petviashvili) equation is derived here describing the propagation of weakly nonlinear and weakly dispersive dust ion acoustic wave in a collisionless unmagnetized plasma consisting of warm adiabatic ions, static negatively charged dust grains, nonthermal electrons, and isothermal positrons. When the coefficient of the nonlinear term of the KP-equation vanishes an appropriate modified KP (MKP) equation describing the propagation of dust ion acoustic wave is derived. Again when the coefficient of the nonlinear term of this MKP equation vanishes, a further modified KP equation is derived. Finally, the stability of the solitary wave solutions of the KPmore » and the different modified KP equations are investigated by the small-k perturbation expansion method of Rowlands and Infeld [J. Plasma Phys. 3, 567 (1969); 8, 105 (1972); 10, 293 (1973); 33, 171 (1985); 41, 139 (1989); Sov. Phys. - JETP 38, 494 (1974)] at the lowest order of k, where k is the wave number of a long-wavelength plane-wave perturbation. The solitary wave solutions of the different evolution equations are found to be stable at this order.« less

A unifying fractional wave equation for compressional and shear waves.

PubMed

Holm, Sverre; Sinkus, Ralph

2010-01-01

This study has been motivated by the observed difference in the range of the power-law attenuation exponent for compressional and shear waves. Usually compressional attenuation increases with frequency to a power between 1 and 2, while shear wave attenuation often is described with powers less than 1. Another motivation is the apparent lack of partial differential equations with desirable properties such as causality that describe such wave propagation. Starting with a constitutive equation which is a generalized Hooke's law with a loss term containing a fractional derivative, one can derive a causal fractional wave equation previously given by Caputo [Geophys J. R. Astron. Soc. 13, 529-539 (1967)] and Wismer [J. Acoust. Soc. Am. 120, 3493-3502 (2006)]. In the low omegatau (low-frequency) case, this equation has an attenuation with a power-law in the range from 1 to 2. This is consistent with, e.g., attenuation in tissue. In the often neglected high omegatau (high-frequency) case, it describes attenuation with a power-law between 0 and 1, consistent with what is observed in, e.g., dynamic elastography. Thus a unifying wave equation derived properly from constitutive equations can describe both cases.

Some new solutions for the Derrida-Lebowitz-Speer-Spohn equation

NASA Astrophysics Data System (ADS)

Ramírez, J.; Romero, J. L.; Tracinà, R.

2013-09-01

The well-known Derrida-Lebowitz-Speer-Spohn equation is investigated. By using specific ansätze and the classical symmetries of the equation, several families of new exact solutions have been found. In particular, there appear traveling waves that include compactons and soliton-compactons. Some other solutions conserve the mass and exhibit diffusion and convection processes from an instantaneous source and localized peakons.

On the dimensionally correct kinetic theory of turbulence for parallel propagation

DOE Office of Scientific and Technical Information (OSTI.GOV)

Gaelzer, R., E-mail: rudi.gaelzer@ufrgs.br, E-mail: yoonp@umd.edu, E-mail: 007gasun@khu.ac.kr, E-mail: luiz.ziebell@ufrgs.br; Ziebell, L. F., E-mail: rudi.gaelzer@ufrgs.br, E-mail: yoonp@umd.edu, E-mail: 007gasun@khu.ac.kr, E-mail: luiz.ziebell@ufrgs.br; Yoon, P. H., E-mail: rudi.gaelzer@ufrgs.br, E-mail: yoonp@umd.edu, E-mail: 007gasun@khu.ac.kr, E-mail: luiz.ziebell@ufrgs.br

2015-03-15

Yoon and Fang [Phys. Plasmas 15, 122312 (2008)] formulated a second-order nonlinear kinetic theory that describes the turbulence propagating in directions parallel/anti-parallel to the ambient magnetic field. Their theory also includes discrete-particle effects, or the effects due to spontaneously emitted thermal fluctuations. However, terms associated with the spontaneous fluctuations in particle and wave kinetic equations in their theory contain proper dimensionality only for an artificial one-dimensional situation. The present paper extends the analysis and re-derives the dimensionally correct kinetic equations for three-dimensional case. The new formalism properly describes the effects of spontaneous fluctuations emitted in three-dimensional space, while the collectivelymore » emitted turbulence propagates predominantly in directions parallel/anti-parallel to the ambient magnetic field. As a first step, the present investigation focuses on linear wave-particle interaction terms only. A subsequent paper will include the dimensionally correct nonlinear wave-particle interaction terms.« less

The structure of shock wave in a gas consisting of ideally elastic, rigid spherical molecules

NASA Technical Reports Server (NTRS)

Cheremisin, F. G.

1972-01-01

Principal approaches are examined to the theoretical study of the shock layer structure. The choice of a molecular model is discussed and three procedures are formulated. These include a numerical calculation method, solution of the kinetic relaxation equation, and solution of the Boltzmann equation.

Separated spin-up and spin-down quantum hydrodynamics of degenerated electrons: Spin-electron acoustic wave appearance.

PubMed

Andreev, Pavel A

2015-03-01

The quantum hydrodynamic (QHD) model of charged spin-1/2 particles contains physical quantities defined for all particles of a species including particles with spin-up and with spin-down. Different populations of states with different spin directions are included in the spin density (the magnetization). In this paper I derive a QHD model, which separately describes spin-up electrons and spin-down electrons. Hence electrons with different projections of spins on the preferable direction are considered as two different species of particles. It is shown that the numbers of particles with different spin directions do not conserve. Hence the continuity equations contain sources of particles. These sources are caused by the interactions of the spins with the magnetic field. Terms of similar nature arise in the Euler equation. The z projection of the spin density is no longer an independent variable. It is proportional to the difference between the concentrations of the electrons with spin-up and the electrons with spin-down. The propagation of waves in the magnetized plasmas of degenerate electrons is considered. Two regimes for the ion dynamics, the motionless ions and the motion of the degenerate ions as the single species with no account of the spin dynamics, are considered. It is shown that this form of the QHD equations gives all solutions obtained from the traditional form of QHD equations with no distinction of spin-up and spin-down states. But it also reveals a soundlike solution called the spin-electron acoustic wave. Coincidence of most solutions is expected since this derivation was started with the same basic equation: the Pauli equation. Solutions arise due to the different Fermi pressures for the spin-up electrons and the spin-down electrons in the magnetic field. The results are applied to degenerate electron gas of paramagnetic and ferromagnetic metals in the external magnetic field. The dispersion of the spin-electron acoustic waves in the partially spin-polarized degenerate neutron matter are also considered.

A double expansion method for the frequency response of finite-length beams with periodic parameters

NASA Astrophysics Data System (ADS)

Ying, Z. G.; Ni, Y. Q.

2017-03-01

A double expansion method for the frequency response of finite-length beams with periodic distribution parameters is proposed. The vibration response of the beam with spatial periodic parameters under harmonic excitations is studied. The frequency response of the periodic beam is the function of parametric period and then can be expressed by the series with the product of periodic and non-periodic functions. The procedure of the double expansion method includes the following two main steps: first, the frequency response function and periodic parameters are expanded by using identical periodic functions based on the extension of the Floquet-Bloch theorem, and the period-parametric differential equation for the frequency response is converted into a series of linear differential equations with constant coefficients; second, the solutions to the linear differential equations are expanded by using modal functions which satisfy the boundary conditions, and the linear differential equations are converted into algebraic equations according to the Galerkin method. The expansion coefficients are obtained by solving the algebraic equations and then the frequency response function is finally determined. The proposed double expansion method can uncouple the effects of the periodic expansion and modal expansion so that the expansion terms are determined respectively. The modal number considered in the second expansion can be reduced remarkably in comparison with the direct expansion method. The proposed double expansion method can be extended and applied to the other structures with periodic distribution parameters for dynamics analysis. Numerical results on the frequency response of the finite-length periodic beam with various parametric wave numbers and wave amplitude ratios are given to illustrate the effective application of the proposed method and the new frequency response characteristics, including the parameter-excited modal resonance, doubling-peak frequency response and remarkable reduction of the maximum frequency response for certain parametric wave number and wave amplitude. The results have the potential application to structural vibration control.

Dust acoustic cnoidal waves in a polytropic complex plasma

NASA Astrophysics Data System (ADS)

El-Labany, S. K.; El-Taibany, W. F.; Abdelghany, A. M.

2018-01-01

The nonlinear characteristics of dust acoustic (DA) waves in an unmagnetized collisionless complex plasma containing adiabatic electrons and ions and negatively charged dust grains (including the effects of modified polarization force) are investigated. Employing the reductive perturbation technique, a Korteweg-de Vries-Burgers (KdVB) equation is derived. The analytical solution for the KdVB equation is discussed. Also, the bifurcation and phase portrait analyses are presented to recognize different types of possible solutions. The dependence of the properties of nonlinear DA waves on the system parameters is investigated. It has been shown that an increase in the value of the modified polarization parameter leads to a fast decay and diminishes the oscillation amplitude of the DA damped cnoidal wave. The relevance of our findings and their possible applications to laboratory and space plasma situations is discussed.

Electromagnetic banana kinetic equation and its applications in tokamaks

NASA Astrophysics Data System (ADS)

Shaing, K. C.; Chu, M. S.; Sabbagh, S. A.; Seol, J.

2018-03-01

A banana kinetic equation in tokamaks that includes effects of the finite banana width is derived for the electromagnetic waves with frequencies lower than the gyro-frequency and the bounce frequency of the trapped particles. The radial wavelengths are assumed to be either comparable to or shorter than the banana width, but much wider than the gyro-radius. One of the consequences of the banana kinetics is that the parallel component of the vector potential is not annihilated by the orbit averaging process and appears in the banana kinetic equation. The equation is solved to calculate the neoclassical quasilinear transport fluxes in the superbanana plateau regime caused by electromagnetic waves. The transport fluxes can be used to model electromagnetic wave and the chaotic magnetic field induced thermal particle or energetic alpha particle losses in tokamaks. It is shown that the parallel component of the vector potential enhances losses when it is the sole transport mechanism. In particular, the fact that the drift resonance can cause significant transport losses in the chaotic magnetic field in the hitherto unknown low collisionality regimes is emphasized.

The Extended Parabolic Equation Method and Implication of Results for Atmospheric Millimeter-Wave and Optical Propagation

NASA Technical Reports Server (NTRS)

Manning, Robert M.

2004-01-01

The extended wide-angle parabolic wave equation applied to electromagnetic wave propagation in random media is considered. A general operator equation is derived which gives the statistical moments of an electric field of a propagating wave. This expression is used to obtain the first and second order moments of the wave field and solutions are found that transcend those which incorporate the full paraxial approximation at the outset. Although these equations can be applied to any propagation scenario that satisfies the conditions of application of the extended parabolic wave equation, the example of propagation through atmospheric turbulence is used. It is shown that in the case of atmospheric wave propagation and under the Markov approximation (i.e., the -correlation of the fluctuations in the direction of propagation), the usual parabolic equation in the paraxial approximation is accurate even at millimeter wavelengths. The methodology developed here can be applied to any qualifying situation involving random propagation through turbid or plasma environments that can be represented by a spectral density of permittivity fluctuations.

Dust acoustic solitary waves in a dusty plasma with two kinds of nonthermal ions at different temperatures

DOE Office of Scientific and Technical Information (OSTI.GOV)

Dorranian, Davoud; Sabetkar, Akbar

The nonlinear dust acoustic solitary waves in a dusty plasma with two nonthermal ion species at different temperatures is studied analytically. Using reductive perturbation method, the Kadomtsev-Petviashivili (KP) equation is derived, and the effects of nonthermal coefficient, ions temperature, and ions number density on the amplitude and width of soliton in dusty plasma are investigated. It is shown that the amplitude of solitary wave of KP equation diverges at critical points of plasma parameters. The modified KP equation is also derived, and from there, the soliton like solutions of modified KP equation with finite amplitude is extracted. Results show thatmore » generation of rarefactive or compressive solitary waves strongly depends on the number and temperature of nonthermal ions. Results of KP equation confirm that for different magnitudes of ions temperature (mass) and number density, mostly compressive solitary waves are generated in a dusty plasma. In this case, the amplitude of solitary wave is decreased, while the width of solitary waves is increased. According to the results of modified KP equation for some certain magnitudes of parameters, there is a condition for generation of an evanescent solitary wave in a dusty plasma.« less

Evolution of nonlinear waves in a blood-filled artery with an aneurysm

NASA Astrophysics Data System (ADS)

Nikolova, E. V.; Jordanov, I. P.; Dimitrova, Z. I.; Vitanov, N. K.

2017-10-01

We discuss propagation of traveling waves in a blood-filled hyper-elastic artery with a local dilatation (an aneurysm). The processes in the injured artery are modeled by an equation of the motion of the arterial wall and by equations of the motion of the fluid (the blood). Taking into account the specific arterial geometry and applying the reductive perturbation method in long-wave approximation we reduce the model equations to a version of the perturbed Korteweg-de Vries kind equation with variable coefficients. Exact traveling-wave solutions of this equation are obtained by the modified method of simplest equation where the differential equation of Abel is used as a simplest equation. A particular case of the obtained exact solution is numerically simulated and discussed from the point of view of arterial disease mechanics.

Multi-Periodic Waves in Shallow Water

DTIC Science & Technology

1992-09-01

models-the Kadomtsev - Petviashvili (KP) equation . The KP equation describes the evolu- tion of weakly nonlinear, weakly two-dimensional waves on water of...experimentally. The analytical model is a family of periodic solutions of the Kadomtsev -Petviashuili equation . The experiments demonstrate the accuracy... Petviashvili Equation (with Norman Schef- fner & Harvey Segur). Proceedings, Nonlinear Water Waves Workshop, University of Bristol. England, 1991. Resonant

From the paddle to the beach - A Boussinesq shallow water numerical wave tank based on Madsen and Sørensen's equations

NASA Astrophysics Data System (ADS)

Orszaghova, Jana; Borthwick, Alistair G. L.; Taylor, Paul H.

2012-01-01

This article describes a one-dimensional numerical model of a shallow-water flume with an in-built piston paddle moving boundary wavemaker. The model is based on a set of enhanced Boussinesq equations and the nonlinear shallow water equations. Wave breaking is described approximately, by locally switching to the nonlinear shallow water equations when a critical wave steepness is reached. The moving shoreline is calculated as part of the solution. The piston paddle wavemaker operates on a movable grid, which is Lagrangian on the paddle face and Eulerian away from the paddle. The governing equations are, however, evolved on a fixed mapped grid, and the newly calculated solution is transformed back onto the moving grid via a domain mapping technique. Validation test results are compared against analytical solutions, confirming correct discretisation of the governing equations, wave generation via the numerical paddle, and movement of the wet/dry front. Simulations are presented that reproduce laboratory experiments of wave runup on a plane beach and wave overtopping of a laboratory seawall, involving solitary waves and compact wave groups. In practice, the numerical model is suitable for simulating the propagation of weakly dispersive waves and can additionally model any associated inundation, overtopping or inland flooding within the same simulation.

Simple equations guide high-frequency surface-wave investigation techniques

USGS Publications Warehouse

Xia, J.; Xu, Y.; Chen, C.; Kaufmann, R.D.; Luo, Y.

2006-01-01

We discuss five useful equations related to high-frequency surface-wave techniques and their implications in practice. These equations are theoretical results from published literature regarding source selection, data-acquisition parameters, resolution of a dispersion curve image in the frequency-velocity domain, and the cut-off frequency of high modes. The first equation suggests Rayleigh waves appear in the shortest offset when a source is located on the ground surface, which supports our observations that surface impact sources are the best source for surface-wave techniques. The second and third equations, based on the layered earth model, reveal a relationship between the optimal nearest offset in Rayleigh-wave data acquisition and seismic setting - the observed maximum and minimum phase velocities, and the maximum wavelength. Comparison among data acquired with different offsets at one test site confirms the better data were acquired with the suggested optimal nearest offset. The fourth equation illustrates that resolution of a dispersion curve image at a given frequency is directly proportional to the product of a length of a geophone array and the frequency. We used real-world data to verify the fourth equation. The last equation shows that the cut-off frequency of high modes of Love waves for a two-layer model is determined by shear-wave velocities and the thickness of the top layer. We applied this equation to Rayleigh waves and multi-layer models with the average velocity and obtained encouraging results. This equation not only endows with a criterion to distinguish high modes from numerical artifacts but also provides a straightforward means to resolve the depth to the half space of a layered earth model. ?? 2005 Elsevier Ltd. All rights reserved.

Pure quasi-P-wave calculation in transversely isotropic media using a hybrid method

NASA Astrophysics Data System (ADS)

Wu, Zedong; Liu, Hongwei; Alkhalifah, Tariq

2018-07-01

The acoustic approximation for anisotropic media is widely used in current industry imaging and inversion algorithms mainly because Pwaves constitute the majority of the energy recorded in seismic exploration. The resulting acoustic formulae tend to be simpler, resulting in more efficient implementations, and depend on fewer medium parameters. However, conventional solutions of the acoustic wave equation with higher-order derivatives suffer from shear wave artefacts. Thus, we derive a new acoustic wave equation for wave propagation in transversely isotropic (TI) media, which is based on a partially separable approximation of the dispersion relation for TI media and free of shear wave artefacts. Even though our resulting equation is not a partial differential equation, it is still a linear equation. Thus, we propose to implement this equation efficiently by combining the finite difference approximation with spectral evaluation of the space-independent parts. The resulting algorithm provides solutions without the constraint ɛ ≥ δ. Numerical tests demonstrate the effectiveness of the approach.

Integrability and Linear Stability of Nonlinear Waves

NASA Astrophysics Data System (ADS)

Degasperis, Antonio; Lombardo, Sara; Sommacal, Matteo

2018-03-01

It is well known that the linear stability of solutions of 1+1 partial differential equations which are integrable can be very efficiently investigated by means of spectral methods. We present here a direct construction of the eigenmodes of the linearized equation which makes use only of the associated Lax pair with no reference to spectral data and boundary conditions. This local construction is given in the general N× N matrix scheme so as to be applicable to a large class of integrable equations, including the multicomponent nonlinear Schrödinger system and the multiwave resonant interaction system. The analytical and numerical computations involved in this general approach are detailed as an example for N=3 for the particular system of two coupled nonlinear Schrödinger equations in the defocusing, focusing and mixed regimes. The instabilities of the continuous wave solutions are fully discussed in the entire parameter space of their amplitudes and wave numbers. By defining and computing the spectrum in the complex plane of the spectral variable, the eigenfrequencies are explicitly expressed. According to their topological properties, the complete classification of these spectra in the parameter space is presented and graphically displayed. The continuous wave solutions are linearly unstable for a generic choice of the coupling constants.

Theoretical relationship between elastic wave velocity and electrical resistivity

NASA Astrophysics Data System (ADS)

Lee, Jong-Sub; Yoon, Hyung-Koo

2015-05-01

Elastic wave velocity and electrical resistivity have been commonly applied to estimate stratum structures and obtain subsurface soil design parameters. Both elastic wave velocity and electrical resistivity are related to the void ratio; the objective of this study is therefore to suggest a theoretical relationship between the two physical parameters. Gassmann theory and Archie's equation are applied to propose a new theoretical equation, which relates the compressional wave velocity to shear wave velocity and electrical resistivity. The piezo disk element (PDE) and bender element (BE) are used to measure the compressional and shear wave velocities, respectively. In addition, the electrical resistivity is obtained by using the electrical resistivity probe (ERP). The elastic wave velocity and electrical resistivity are recorded in several types of soils including sand, silty sand, silty clay, silt, and clay-sand mixture. The appropriate input parameters are determined based on the error norm in order to increase the reliability of the proposed relationship. The predicted compressional wave velocities from the shear wave velocity and electrical resistivity are similar to the measured compressional velocities. This study demonstrates that the new theoretical relationship may be effectively used to predict the unknown geophysical property from the measured values.

On the three dimensional structure of stratospheric material transport associated with various types of waves

NASA Astrophysics Data System (ADS)

Kinoshita, T.; Sato, K.

2016-12-01

The Transformed Eulerian-Mean (TEM) equations were derived by Andrews and McIntyre (1976, 1978) and have been widely used to examine wave-mean flow interaction in the meridional cross section. According to previous studies, the Brewer-Dobson circulation in the stratosphere is driven by planetary waves, baroclinic waves, and inertia-gravity waves, and that the meridional circulation from the summer hemisphere to the winter hemisphere in the mesosphere is mainly driven by gravity waves (e.g., Garcia and Boville 1994; Plumb and Semeniuk 2003; Watanabe et al. 2008; Okamoto et al. 2011). However, the TEM equations do not provide the three-dimensional view of the transport, so that the three dimensional TEM equations have been formulated (Hoskins et al. 1983, Trenberth 1986, Plumb 1985, 1986, Takaya and Nakamura 1997, 2001, Miyahara 2006, Kinoshita et al. 2010, Noda 2010, Kinoshita and Sato 2013a, b, and Noda 2014). On the other hand, the TEM equations cannot properly treat the lower boundary and unstable waves. The Mass-weighted Isentropic Mean (MIM) equations derived by Iwasaki (1989, 1990) are the equations that overcome those problems and the formulation of three-dimensional MIM equations have been studied. The present study applies the three-dimensional TEM and MIM equations to the ERA-Interim reanalysis data and examines the climatological character of three-dimensional structure of Stratospheric Brewer-Dobson circulation. Next, we will discuss how to treat the flow associated with spatial structure of stationary waves.

Nonlinear Drift-Kinetic Equation in the Presence of a Circularly Polarized Wave

NASA Technical Reports Server (NTRS)

Khazanov, G. V.; Krivorutsky, E. N.; Whitaker, Ann F. (Technical Monitor)

2001-01-01

Equations of the single particle motion and nonlinear kinetic equation for plasma in the presence of a circularly polarized wave of arbitrary frequency in the drift approximation are presented. The nonstationarity and inhomogeneity of the plasma-wave system are taken into account.

On the exact solutions of high order wave equations of KdV type (I)

NASA Astrophysics Data System (ADS)

Bulut, Hasan; Pandir, Yusuf; Baskonus, Haci Mehmet

2014-12-01

In this paper, by means of a proper transformation and symbolic computation, we study high order wave equations of KdV type (I). We obtained classification of exact solutions that contain soliton, rational, trigonometric and elliptic function solutions by using the extended trial equation method. As a result, the motivation of this paper is to utilize the extended trial equation method to explore new solutions of high order wave equation of KdV type (I). This method is confirmed by applying it to this kind of selected nonlinear equations.

Solitary wave solutions of two-dimensional nonlinear Kadomtsev-Petviashvili dynamic equation in dust-acoustic plasmas

NASA Astrophysics Data System (ADS)

Seadawy, Aly R.

2017-09-01

Nonlinear two-dimensional Kadomtsev-Petviashvili (KP) equation governs the behaviour of nonlinear waves in dusty plasmas with variable dust charge and two temperature ions. By using the reductive perturbation method, the two-dimensional dust-acoustic solitary waves (DASWs) in unmagnetized cold plasma consisting of dust fluid, ions and electrons lead to a KP equation. We derived the solitary travelling wave solutions of the two-dimensional nonlinear KP equation by implementing sech-tanh, sinh-cosh, extended direct algebraic and fraction direct algebraic methods. We found the electrostatic field potential and electric field in the form travelling wave solutions for two-dimensional nonlinear KP equation. The solutions for the KP equation obtained by using these methods can be demonstrated precisely and efficiency. As an illustration, we used the readymade package of Mathematica program 10.1 to solve the original problem. These solutions are in good agreement with the analytical one.

Calculation of the Full Scattering Amplitude without Partial Wave Decomposition II: Inclusion of Exchange

NASA Technical Reports Server (NTRS)

Shertzer, Janine; Temkin, A.

2003-01-01

As is well known, the full scattering amplitude can be expressed as an integral involving the complete scattering wave function. We have shown that the integral can be simplified and used in a practical way. Initial application to electron-hydrogen scattering without exchange was highly successful. The Schrodinger equation (SE), which can be reduced to a 2d partial differential equation (pde), was solved using the finite element method. We have now included exchange by solving the resultant SE, in the static exchange approximation, which is reducible to a pair of coupled pde's. The resultant scattering amplitudes, both singlet and triplet, calculated as a function of energy are in excellent agreement with converged partial wave results.

Wave propagation through an inhomogeneous slab sandwiched by the piezoelectric and the piezomagnetic half spaces.

PubMed

Jiao, Fengyu; Wei, Peijun; Li, Li

2017-01-01

Wave propagation through a gradient slab sandwiched by the piezoelectric and the piezomagnetic half spaces are studied in this paper. First, the secular equations in the transverse isotropic piezoelectric/piezomagnetic half spaces are derived from the general dynamic equation. Then, the state vectors at piezoelectric and piezomagnetic half spaces are related to the amplitudes of various possible waves. The state transfer equation of the functionally graded slab is derived from the equations of motion by the reduction of order, and the transfer matrix of the functionally gradient slab is obtained by solving the state transfer equation with the spatial-varying coefficient. Finally, the continuous interface conditions are used to lead to the resultant algebraic equations. The algebraic equations are solved to obtain the amplitude ratios of various waves which are further used to obtain the energy reflection and transmission coefficients of various waves. The numerical results are shown graphically and are validated by the energy conservation law. Based on the numerical results on the fives of gradient profiles, the influences of the graded slab on the wave propagation are discussed. It is found that the reflection and transmission coefficients are obviously dependent upon the gradient profile. The various surface waves are more sensitive to the gradient profile than the bulk waves. Copyright © 2016 Elsevier B.V. All rights reserved.

The limitation and applicability of Musher-Sturman equation to two dimensional lower hybrid wave collapse

NASA Technical Reports Server (NTRS)

Tam, Sunny W. Y.; Chang, Tom

1995-01-01

The existence of localized regions of intense lower hybrid waves in the auroral ionosphere recently observed by rocket and satellite experiments can be understood by the study of a non-linear two-timescale coupling process. In this Letter, we demonstrate that the leading non-linear term in the standard Musher-Sturman equation vanishes identically in strict two-dimensions (normal to the magnetic field). Instead, the new two-dimensional equation is characterized by a much weaker non-linear term which arises from the ponderomotive force perpendicular to the magnetic field, particularly that due to the ions. The old and new equations are compared by means of time-evolution calculations of wave fields. The results exhibit a remarkable difference in the evolution of the waves as governed by the two equations. Such dissimilar outcomes motivate our investigation of the limitation of Musher-Sturman equation in quasi-two-dimensions. Only within all these limits can Musher-Sturman equation adequately describe the collapse of lower hybrid waves.

Computational procedures for mixed equations with shock waves

NASA Technical Reports Server (NTRS)

Yu, N. J.; Seebass, R.

1974-01-01

This paper discusses the procedures we have developed to treat a canonical problem involving a mixed nonlinear equation with boundary data that imply a discontinuous solution. This equation arises in various physical contexts and is basic to the description of the nonlinear acoustic behavior of a shock wave near a caustic. The numerical scheme developed is of second order, treats discontinuities as such by applying the appropriate jump conditions across them, and eliminates the numerical dissipation and dispersion associated with large gradients. Our results are compared with the results of a first-order scheme and with those of a second-order scheme we have developed. The algorithm used here can easily be generalized to more complicated problems, including transonic flows with imbedded shocks.

Comment on “Approximate solutions of the Dirac equation for the Rosen-Morse potential including the spin-orbit centrifugal term” [J. Math. Phys. 51, 023525 (2010)

DOE Office of Scientific and Technical Information (OSTI.GOV)

Ghoumaid, A.; Benamira, F.; Guechi, L.

2016-02-15

It is shown that the application of the Nikiforov-Uvarov method by Ikhdair for solving the Dirac equation with the radial Rosen-Morse potential plus the spin-orbit centrifugal term is inadequate because the required conditions are not satisfied. The energy spectra given is incorrect and the wave functions are not physically acceptable. We clarify the problem and prove that the spinor wave functions are expressed in terms of the generalized hypergeometric functions {sub 2}F{sub 1}(a, b, c; z). The energy eigenvalues for the bound states are given by the solution of a transcendental equation involving the hypergeometric function.

Multi-hump potentials for efficient wave absorption in the numerical solution of the time-dependent Schrödinger equation

NASA Astrophysics Data System (ADS)

Silaev, A. A.; Romanov, A. A.; Vvedenskii, N. V.

2018-03-01

In the numerical solution of the time-dependent Schrödinger equation by grid methods, an important problem is the reflection and wrap-around of the wave packets at the grid boundaries. Non-optimal absorption of the wave function leads to possible large artifacts in the results of numerical simulations. We propose a new method for the construction of the complex absorbing potentials for wave suppression at the grid boundaries. The method is based on the use of the multi-hump imaginary potential which contains a sequence of smooth and symmetric humps whose widths and amplitudes are optimized for wave absorption in different spectral intervals. We show that this can ensure a high efficiency of absorption in a wide range of de Broglie wavelengths, which includes wavelengths comparable to the width of the absorbing layer. Therefore, this method can be used for high-precision simulations of various phenomena where strong spreading of the wave function takes place, including the phenomena accompanying the interaction of strong fields with atoms and molecules. The efficiency of the proposed method is demonstrated in the calculation of the spectrum of high-order harmonics generated during the interaction of hydrogen atoms with an intense infrared laser pulse.

Existence and Stability of Traveling Waves for Degenerate Reaction-Diffusion Equation with Time Delay

NASA Astrophysics Data System (ADS)

Huang, Rui; Jin, Chunhua; Mei, Ming; Yin, Jingxue

2018-01-01

This paper deals with the existence and stability of traveling wave solutions for a degenerate reaction-diffusion equation with time delay. The degeneracy of spatial diffusion together with the effect of time delay causes us the essential difficulty for the existence of the traveling waves and their stabilities. In order to treat this case, we first show the existence of smooth- and sharp-type traveling wave solutions in the case of c≥c^* for the degenerate reaction-diffusion equation without delay, where c^*>0 is the critical wave speed of smooth traveling waves. Then, as a small perturbation, we obtain the existence of the smooth non-critical traveling waves for the degenerate diffusion equation with small time delay τ >0 . Furthermore, we prove the global existence and uniqueness of C^{α ,β } -solution to the time-delayed degenerate reaction-diffusion equation via compactness analysis. Finally, by the weighted energy method, we prove that the smooth non-critical traveling wave is globally stable in the weighted L^1 -space. The exponential convergence rate is also derived.

Existence and Stability of Traveling Waves for Degenerate Reaction-Diffusion Equation with Time Delay

NASA Astrophysics Data System (ADS)

Huang, Rui; Jin, Chunhua; Mei, Ming; Yin, Jingxue

2018-06-01

This paper deals with the existence and stability of traveling wave solutions for a degenerate reaction-diffusion equation with time delay. The degeneracy of spatial diffusion together with the effect of time delay causes us the essential difficulty for the existence of the traveling waves and their stabilities. In order to treat this case, we first show the existence of smooth- and sharp-type traveling wave solutions in the case of c≥c^* for the degenerate reaction-diffusion equation without delay, where c^*>0 is the critical wave speed of smooth traveling waves. Then, as a small perturbation, we obtain the existence of the smooth non-critical traveling waves for the degenerate diffusion equation with small time delay τ >0. Furthermore, we prove the global existence and uniqueness of C^{α ,β }-solution to the time-delayed degenerate reaction-diffusion equation via compactness analysis. Finally, by the weighted energy method, we prove that the smooth non-critical traveling wave is globally stable in the weighted L^1-space. The exponential convergence rate is also derived.

Modeling electromagnetic ion cyclotron waves in the inner magnetosphere

NASA Astrophysics Data System (ADS)

Gamayunov, Konstantin; Engebretson, Mark; Zhang, Ming; Rassoul, Hamid

The evolution of He+-mode electromagnetic ion cyclotron (EMIC) waves is studied inside the geostationary orbit using our global model of ring current (RC) ions, electric field, plasmasphere, and EMIC waves. In contrast to the approach previously used by Gamayunov et al. [2009], however, we do not use the bounce-averaged wave kinetic equation but instead use a complete, non bounce-averaged, equation to model the evolution of EMIC wave power spectral density, including off-equatorial wave dynamics. The major results of our study can be summarized as follows. (1) The thermal background level for EMIC waves is too low to allow waves to grow up to the observable level during one pass between the “bi-ion latitudes” (the latitudes where the given wave frequency is equal to the O+-He+ bi-ion frequency) in conjugate hemispheres. As a consequence, quasi-field-aligned EMIC waves are not typically produced in the model if the thermal background level is used, but routinely observed in the Earth’s magnetosphere. To overcome this model-observation discrepancy we suggest a nonlinear energy cascade from the lower frequency range of ultra low frequency waves into the frequency range of EMIC wave generation as a possible mechanism supplying the needed level of seed fluctuations that guarantees growth of EMIC waves during one pass through the near equatorial region. The EMIC wave development from a suprathermal background level shows that EMIC waves are quasi-field-aligned near the equator, while they are oblique at high latitudes, and the Poynting flux is predominantly directed away from the near equatorial source region in agreement with observations. (2) An abundance of O+ strongly controls the energy of oblique He+-mode EMIC waves that propagate to the equator after their reflection at “bi-ion latitudes”, and so it controls a fraction of wave energy in the oblique normals. (3) The RC O+ not only causes damping of the He+-mode EMIC waves but also causes wave generation in the region of highly oblique wave normal angles, typically for theta > 82deg, where a growth rate gamma > 0.01 rad/s is frequently observed. The instability is driven by the loss-cone feature in the RC O+ distribution function. (4) The oblique and intense He+-mode EMIC waves generated by RC O+ in the region L ˜ 2-3 may have an implication to the energetic particle loss in the inner radiation belt. Acknowledgments: This paper is based upon work supported by the National Science Foundation under Grant Number AGS-1203516.

Formation of wave packets in the Ostrovsky equation for both normal and anomalous dispersion

PubMed Central

Grimshaw, Roger; Stepanyants, Yury; Alias, Azwani

2016-01-01

It is well known that the Ostrovsky equation with normal dispersion does not support steady solitary waves. An initial Korteweg–de Vries solitary wave decays adiabatically through the radiation of long waves and is eventually replaced by an envelope solitary wave whose carrier wave and envelope move with different velocities (phase and group velocities correspondingly). Here, we examine the same initial condition for the Ostrovsky equation with anomalous dispersion, when the wave frequency increases with wavenumber in the limit of very short waves. The essential difference is that now there exists a steady solitary wave solution (Ostrovsky soliton), which in the small-amplitude limit can be described asymptotically through the solitary wave solution of a nonlinear Schrödinger equation, based at that wavenumber where the phase and group velocities coincide. Long-time numerical simulations show that the emergence of this steady envelope solitary wave is a very robust feature. The initial Korteweg–de Vries solitary wave transforms rapidly to this envelope solitary wave in a seemingly non-adiabatic manner. The amplitude of the Ostrovsky soliton strongly correlates with the initial Korteweg–de Vries solitary wave. PMID:26997887

Well-posed two-temperature constitutive equations for stable dense fluid shock waves using molecular dynamics and generalizations of Navier-Stokes-Fourier continuum mechanics.

PubMed

Hoover, Wm G; Hoover, Carol G

2010-04-01

Guided by molecular dynamics simulations, we generalize the Navier-Stokes-Fourier constitutive equations and the continuum motion equations to include both transverse and longitudinal temperatures. To do so we partition the contributions of the heat transfer, the work done, and the heat flux vector between the longitudinal and transverse temperatures. With shockwave boundary conditions time-dependent solutions of these equations converge to give stationary shockwave profiles. The profiles include anisotropic temperature and can be fitted to molecular dynamics results, demonstrating the utility and simplicity of a two-temperature description of far-from-equilibrium states.

Maintenance of Austral Summertime Upper-Tropospheric Circulation over Tropical South America: The Bolivian High-Nordeste Low System.

NASA Astrophysics Data System (ADS)

Chen, Tsing-Chang; Weng, Shu-Ping; Schubert, Siegfried

1999-07-01

Using the NASA/GEOS reanalysis data for 1980-95, the austral-summer stationary eddies in the tropical-subtropical Southern Hemisphere are examined in two wave regimes: long and short wave (wave 1 and waves 2-6, respectively). The basic structure of the Bolivian high-Nordeste low (BH-NL) system is formed by a short-wave train across South America but modulated by the long-wave regime. The short-wave train exhibits a monsoonlike vertical phase reversal in the midtroposphere and a quarter-wave phase shift relative to the divergent circulation. As inferred from (a) the spatial relationship between the streamfunction and velocity potential and (b) the structure of the divergent circulation, the short-wave train forming the BH-NL system is maintained by South American local heating and remote African heating, while the long-wave regime is maintained by western tropical Pacific heating.The maintenance of the stationary waves in the two wave regimes is further illustrated by a simple diagnostic scheme that includes the velocity-potential maintenance equation (which links velocity potential and diabatic heating) and the streamfunction budget (which is the inverse Laplacian transform of the vorticity equation). Some simple relationships between streamfunction and velocity potential for both wave regimes are established to substantiate the links between diabatic heating and streamfunction; of particular interest is a Sverdrup balance in the short-wave regime. This simplified vorticity equation explains the vertical structure of the short-wave train associated with the BH-NL system and its spatial relationship with the divergent circulation.Based upon the diagnostic analysis of its maintenance a simple forced barotropic model is adopted to simulate the BH-NL system with idealized forcings, which imitates the real 200-mb divergence centers over South America, Africa, and the tropical Pacific. Numerical simulations demonstrate that the formation of the BH-NL system is affected not only by the African remote forcing, but also by the tropical Pacific forcing.

Spatiotemporal optical dark X solitary waves.

PubMed

Baronio, Fabio; Chen, Shihua; Onorato, Miguel; Trillo, Stefano; Wabnitz, Stefan; Kodama, Yuji

2016-12-01

We introduce spatiotemporal optical dark X solitary waves of the (2+1)D hyperbolic nonlinear Schrödinger equation (NLSE), which rules wave propagation in a self-focusing and normally dispersive medium. These analytical solutions are derived by exploiting the connection between the NLSE and a well-known equation of hydrodynamics, namely the type II Kadomtsev-Petviashvili (KP-II) equation. As a result, families of shallow water X soliton solutions of the KP-II equation are mapped into optical dark X solitary wave solutions of the NLSE. Numerical simulations show that optical dark X solitary waves may propagate for long distances (tens of nonlinear lengths) before they eventually break up, owing to the modulation instability of the continuous wave background. This finding opens a novel path for the excitation and control of X solitary waves in nonlinear optics.

A Self-Consistent Model of the Interacting Ring Current Ions and Electromagnetic Ion Cyclotron Waves, Initial Results: Waves and Precipitating Fluxes

NASA Technical Reports Server (NTRS)

Khazanov, G. V.; Gamayunov, K. V.; Jordanova, V. K.; Krivorutsky, E. N.

2002-01-01

Initial results from a newly developed model of the interacting ring current ions and ion cyclotron waves are presented. The model is based on the system of two kinetic equations: one equation describes the ring current ion dynamics, and another equation describes wave evolution. The system gives a self-consistent description of the ring current ions and ion cyclotron waves in a quasilinear approach. These equations for the ion phase space distribution function and for the wave power spectral density were solved on aglobal magnetospheric scale undernonsteady state conditions during the 2-5 May 1998 storm. The structure and dynamics of the ring current proton precipitating flux regions and the ion cyclotron wave-active zones during extreme geomagnetic disturbances on 4 May 1998 are presented and discussed in detail.

A pseudoenergy wave-activity relation for ageostrophic and non-hydrostatic moist atmosphere

NASA Astrophysics Data System (ADS)

Ran, Ling-Kun; Ping, Fan

2015-05-01

By employing the energy-Casimir method, a three-dimensional virtual pseudoenergy wave-activity relation for a moist atmosphere is derived from a complete system of nonhydrostatic equations in Cartesian coordinates. Since this system of equations includes the effects of water substance, mass forcing, diabatic heating, and dissipations, the derived wave-activity relation generalizes the previous result for a dry atmosphere. The Casimir function used in the derivation is a monotonous function of virtual potential vorticity and virtual potential temperature. A virtual energy equation is employed (in place of the previous zonal momentum equation) in the derivation, and the basic state is stationary but can be three-dimensional or, at least, not necessarily zonally symmetric. The derived wave-activity relation is further used for the diagnosis of the evolution and propagation of meso-scale weather systems leading to heavy rainfall. Our diagnosis of two real cases of heavy precipitation shows that positive anomalies of the virtual pseudoenergy wave-activity density correspond well with the strong precipitation and are capable of indicating the movement of the precipitation region. This is largely due to the cyclonic vorticity perturbation and the vertically increasing virtual potential temperature over the precipitation region. Project supported by the National Basic Research Program of China (Grant No. 2013CB430105), the Key Program of the Chinese Academy of Sciences (Grant No. KZZD-EW-05), the National Natural Science Foundation of China (Grant No. 41175060), and the Project of CAMS, China (Grant No. 2011LASW-B15).

Alternative Form of the Hydrogenic Wave Functions for an Extended, Uniformly Charged Nucleus.

ERIC Educational Resources Information Center

Ley-Koo, E.; And Others

1980-01-01

Presented are forms of harmonic oscillator attraction and Coulomb wave functions which can be explicitly constructed and which lead to numerical results for the energy eigenvalues and eigenfunctions of the atomic system. The Schrodinger equation and its solution and specific cases of muonic atoms illustrating numerical calculations are included.…

Dirac and Klein-Gordon-Fock equations in Grumiller’s spacetime

NASA Astrophysics Data System (ADS)

Al-Badawi, A.; Sakalli, I.

We study the Dirac and the chargeless Klein-Gordon-Fock equations in the geometry of Grumiller’s spacetime that describes a model for gravity of a central object at large distances. The Dirac equation is separated into radial and angular equations by adopting the Newman-Penrose formalism. The angular part of the both wave equations are analytically solved. For the radial equations, we managed to reduce them to one dimensional Schrödinger-type wave equations with their corresponding effective potentials. Fermions’s potentials are numerically analyzed by serving their some characteristic plots. We also compute the quasinormal frequencies of the chargeless and massive scalar waves. With the aid of those quasinormal frequencies, Bekenstein’s area conjecture is tested for the Grumiller black hole. Thus, the effects of the Rindler acceleration on the waves of fermions and scalars are thoroughly analyzed.

Causes of plasma column contraction in surface-wave-driven discharges in argon at atmospheric pressure

NASA Astrophysics Data System (ADS)

Ridenti, Marco Antonio; de Amorim, Jayr; Dal Pino, Arnaldo; Guerra, Vasco; Petrov, George

2018-01-01

In this work we compute the main features of a surface-wave-driven plasma in argon at atmospheric pressure in view of a better understanding of the contraction phenomenon. We include the detailed chemical kinetics dynamics of Ar and solve the mass conservation equations of the relevant neutral excited and charged species. The gas temperature radial profile is calculated by means of the thermal diffusion equation. The electric field radial profile is calculated directly from the numerical solution of the Maxwell equations assuming the surface wave to be propagating in the TM00 mode. The problem is considered to be radially symmetrical, the axial variations are neglected, and the equations are solved in a self-consistent fashion. We probe the model results considering three scenarios: (i) the electron energy distribution function (EEDF) is calculated by means of the Boltzmann equation; (ii) the EEDF is considered to be Maxwellian; (iii) the dissociative recombination is excluded from the chemical kinetics dynamics, but the nonequilibrium EEDF is preserved. From this analysis, the dissociative recombination is shown to be the leading mechanism in the constriction of surface-wave plasmas. The results are compared with mass spectrometry measurements of the radial density profile of the ions Ar+ and Ar2+. An explanation is proposed for the trends seen by Thomson scattering diagnostics that shows a substantial increase of electron temperature towards the plasma borders where the electron density is small.

Unsteady Solution of Non-Linear Differential Equations Using Walsh Function Series

NASA Technical Reports Server (NTRS)

Gnoffo, Peter A.

2015-01-01

Walsh functions form an orthonormal basis set consisting of square waves. The discontinuous nature of square waves make the system well suited for representing functions with discontinuities. The product of any two Walsh functions is another Walsh function - a feature that can radically change an algorithm for solving non-linear partial differential equations (PDEs). The solution algorithm of non-linear differential equations using Walsh function series is unique in that integrals and derivatives may be computed using simple matrix multiplication of series representations of functions. Solutions to PDEs are derived as functions of wave component amplitude. Three sample problems are presented to illustrate the Walsh function series approach to solving unsteady PDEs. These include an advection equation, a Burgers equation, and a Riemann problem. The sample problems demonstrate the use of the Walsh function solution algorithms, exploiting Fast Walsh Transforms in multi-dimensions (O(Nlog(N))). Details of a Fast Walsh Reciprocal, defined here for the first time, enable inversion of aWalsh Symmetric Matrix in O(Nlog(N)) operations. Walsh functions have been derived using a fractal recursion algorithm and these fractal patterns are observed in the progression of pairs of wave number amplitudes in the solutions. These patterns are most easily observed in a remapping defined as a fractal fingerprint (FFP). A prolongation of existing solutions to the next highest order exploits these patterns. The algorithms presented here are considered a work in progress that provide new alternatives and new insights into the solution of non-linear PDEs.

Dispersion relations of elastic waves in one-dimensional piezoelectric/piezomagnetic phononic crystal with initial stresses.

PubMed

Guo, Xiao; Wei, Peijun

2016-03-01

The dispersion relations of elastic waves in a one-dimensional phononic crystal formed by periodically repeating of a pre-stressed piezoelectric slab and a pre-stressed piezomagnetic slab are studied in this paper. The influences of initial stress on the dispersive relation are considered based on the incremental stress theory. First, the incremental stress theory of elastic solid is extended to the magneto-electro-elasto solid. The governing equations, constitutive equations, and boundary conditions of the incremental stresses in a magneto-electro-elasto solid are derived with consideration of the existence of initial stresses. Then, the transfer matrices of a pre-stressed piezoelectric slab and a pre-stressed piezomagnetic slab are formulated, respectively. The total transfer matrix of a single cell in the phononic crystal is obtained by the multiplication of two transfer matrixes related with two adjacent slabs. Furthermore, the Bloch theorem is used to obtain the dispersive equations of in-plane and anti-plane Bloch waves. The dispersive equations are solved numerically and the numerical results are shown graphically. The oblique propagation and the normal propagation situations are both considered. In the case of normal propagation of elastic waves, the analytical expressions of the dispersion equation are derived and compared with other literatures. The influences of initial stresses, including the normal initial stresses and shear initial stresses, on the dispersive relations are both discussed based on the numerical results. Copyright © 2015 Elsevier B.V. All rights reserved.

Hydrodynamic optical soliton tunneling

NASA Astrophysics Data System (ADS)

Sprenger, P.; Hoefer, M. A.; El, G. A.

2018-03-01

A notion of hydrodynamic optical soliton tunneling is introduced in which a dark soliton is incident upon an evolving, broad potential barrier that arises from an appropriate variation of the input signal. The barriers considered include smooth rarefaction waves and highly oscillatory dispersive shock waves. Both the soliton and the barrier satisfy the same one-dimensional defocusing nonlinear Schrödinger (NLS) equation, which admits a convenient dispersive hydrodynamic interpretation. Under the scale separation assumption of nonlinear wave (Whitham) modulation theory, the highly nontrivial nonlinear interaction between the soliton and the evolving hydrodynamic barrier is described in terms of self-similar, simple wave solutions to an asymptotic reduction of the Whitham-NLS partial differential equations. One of the Riemann invariants of the reduced modulation system determines the characteristics of a soliton interacting with a mean flow that results in soliton tunneling or trapping. Another Riemann invariant yields the tunneled soliton's phase shift due to hydrodynamic interaction. Soliton interaction with hydrodynamic barriers gives rise to effects that include reversal of the soliton propagation direction and spontaneous soliton cavitation, which further suggest possible methods of dark soliton control in optical fibers.

Hydrodynamic optical soliton tunneling.

PubMed

Sprenger, P; Hoefer, M A; El, G A

2018-03-01

A notion of hydrodynamic optical soliton tunneling is introduced in which a dark soliton is incident upon an evolving, broad potential barrier that arises from an appropriate variation of the input signal. The barriers considered include smooth rarefaction waves and highly oscillatory dispersive shock waves. Both the soliton and the barrier satisfy the same one-dimensional defocusing nonlinear Schrödinger (NLS) equation, which admits a convenient dispersive hydrodynamic interpretation. Under the scale separation assumption of nonlinear wave (Whitham) modulation theory, the highly nontrivial nonlinear interaction between the soliton and the evolving hydrodynamic barrier is described in terms of self-similar, simple wave solutions to an asymptotic reduction of the Whitham-NLS partial differential equations. One of the Riemann invariants of the reduced modulation system determines the characteristics of a soliton interacting with a mean flow that results in soliton tunneling or trapping. Another Riemann invariant yields the tunneled soliton's phase shift due to hydrodynamic interaction. Soliton interaction with hydrodynamic barriers gives rise to effects that include reversal of the soliton propagation direction and spontaneous soliton cavitation, which further suggest possible methods of dark soliton control in optical fibers.

Freezing optical rogue waves by Zeno dynamics

NASA Astrophysics Data System (ADS)

Bayındır, Cihan; Ozaydin, Fatih

2018-04-01

We investigate the Zeno dynamics of the optical rogue waves. Considering their usage in modeling rogue wave dynamics, we analyze the Zeno dynamics of the Akhmediev breathers, Peregrine and Akhmediev-Peregrine soliton solutions of the nonlinear Schrödinger equation. We show that frequent measurements of the wave inhibits its movement in the observation domain for each of these solutions. We analyze the spectra of the rogue waves under Zeno dynamics. We also analyze the effect of observation frequency on the rogue wave profile and on the probability of lingering of the wave in the observation domain. Our results can find potential applications in optics including nonlinear phenomena.

Theory of helix traveling wave tubes with dielectric and vane loading

DOE Office of Scientific and Technical Information (OSTI.GOV)

Freund, H.P.; Zaidman, E.G.; Antonsen, T.M. Jr.

1996-08-01

A time-dependent nonlinear analysis of a helix traveling wave tube (TWT) is presented for a configuration where an electron beam propagates through a sheath helix surrounded by a conducting wall. The effects of dielectric and vane loading are included in the formulation as is efficiency enhancement by tapering the helix pitch. Dielectric loading is described under the assumption that the gap between the helix and the wall is uniformly filled by a dielectric material. The vane-loading model describes the insertion of an arbitrary number of vanes running the length of the helix, and the polarization of the field between themore » vanes is assumed to be an azimuthally symmetric transverse-electric mode. The field is represented as a superposition of azimuthally symmetric waves in a vacuum sheath helix. An overall explicit sinusoidal variation of the form exp({ital ikz}{minus}{ital i}{omega}{ital t}) is assumed (where {omega} denotes the angular frequency corresponding to the wave number {ital k} in the vacuum sheath helix), and the polarization and radial variation of each wave is determined by the boundary conditions in a vacuum sheath helix. The propagation of each wave {ital in} {ital vacuo} as well as the interaction of each wave with the electron beam is included by allowing the amplitudes of the waves to vary in {ital z} and {ital t}. A dynamical equation for the field amplitudes is derived analogously to Poynting{close_quote}s equation, and solved in conjunction with the three-dimensional Lorentz force equations for an ensemble of electrons. Electron beams with a both a continuous and emission-gated pulse format are analyzed, and the model is compared with linear theory of the interaction as well as with the performance of a TWTs operated at the Naval Research Laboratory and at Northrop{endash}Grumman Corporation. {copyright} {ital 1996 American Institute of Physics.}« less

Waves and instabilities in plasmas

DOE Office of Scientific and Technical Information (OSTI.GOV)

Chen, L.

1987-01-01

The contents of this book are: Plasma as a Dielectric Medium; Nyquist Technique; Absolute and Convective Instabilities; Landau Damping and Phase Mixing; Particle Trapping and Breakdown of Linear Theory; Solution of Viasov Equation via Guilding-Center Transformation; Kinetic Theory of Magnetohydrodynamic Waves; Geometric Optics; Wave-Kinetic Equation; Cutoff and Resonance; Resonant Absorption; Mode Conversion; Gyrokinetic Equation; Drift Waves; Quasi-Linear Theory; Ponderomotive Force; Parametric Instabilities; Problem Sets for Homework, Midterm and Final Examinations.

An Operator Method for Field Moments from the Extended Parabolic Wave Equation and Analytical Solutions of the First and Second Moments for Atmospheric Electromagnetic Wave Propagation

NASA Technical Reports Server (NTRS)

Manning, Robert M.

2004-01-01

The extended wide-angle parabolic wave equation applied to electromagnetic wave propagation in random media is considered. A general operator equation is derived which gives the statistical moments of an electric field of a propagating wave. This expression is used to obtain the first and second order moments of the wave field and solutions are found that transcend those which incorporate the full paraxial approximation at the outset. Although these equations can be applied to any propagation scenario that satisfies the conditions of application of the extended parabolic wave equation, the example of propagation through atmospheric turbulence is used. It is shown that in the case of atmospheric wave propagation and under the Markov approximation (i.e., the delta-correlation of the fluctuations in the direction of propagation), the usual parabolic equation in the paraxial approximation is accurate even at millimeter wavelengths. The comprehensive operator solution also allows one to obtain expressions for the longitudinal (generalized) second order moment. This is also considered and the solution for the atmospheric case is obtained and discussed. The methodology developed here can be applied to any qualifying situation involving random propagation through turbid or plasma environments that can be represented by a spectral density of permittivity fluctuations.

The Shock and Vibration Digest. Volume 16, Number 11

DTIC Science & Technology

1984-11-01

wave [19], a secular equation for Rayleigh waves on ing, seismic risk, and related problems are discussed. the surface of an anisotropic half-space...waves in an !so- tive equation of an elastic-plastic rack medium was....... tropic linear elastic half-space with plane material used; the coefficient...pair of semi-linear hyperbolic partial differential -- " Conditions under which the equations of motion equations governing slow variations in amplitude

Nonlinear and linear wave equations for propagation in media with frequency power law losses

NASA Astrophysics Data System (ADS)

Szabo, Thomas L.

2003-10-01

The Burgers, KZK, and Westervelt wave equations used for simulating wave propagation in nonlinear media are based on absorption that has a quadratic dependence on frequency. Unfortunately, most lossy media, such as tissue, follow a more general frequency power law. The authors first research involved measurements of loss and dispersion associated with a modification to Blackstock's solution to the linear thermoviscous wave equation [J. Acoust. Soc. Am. 41, 1312 (1967)]. A second paper by Blackstock [J. Acoust. Soc. Am. 77, 2050 (1985)] showed the loss term in the Burgers equation for plane waves could be modified for other known instances of loss. The authors' work eventually led to comprehensive time-domain convolutional operators that accounted for both dispersion and general frequency power law absorption [Szabo, J. Acoust. Soc. Am. 96, 491 (1994)]. Versions of appropriate loss terms were developed to extend the standard three nonlinear wave equations to these more general losses. Extensive experimental data has verified the predicted phase velocity dispersion for different power exponents for the linear case. Other groups are now working on methods suitable for solving wave equations numerically for these types of loss directly in the time domain for both linear and nonlinear media.

Fast and local non-linear evolution of steep wave-groups on deep water: A comparison of approximate models to fully non-linear simulations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Adcock, T. A. A.; Taylor, P. H.

2016-01-15

The non-linear Schrödinger equation and its higher order extensions are routinely used for analysis of extreme ocean waves. This paper compares the evolution of individual wave-packets modelled using non-linear Schrödinger type equations with packets modelled using fully non-linear potential flow models. The modified non-linear Schrödinger Equation accurately models the relatively large scale non-linear changes to the shape of wave-groups, with a dramatic contraction of the group along the mean propagation direction and a corresponding extension of the width of the wave-crests. In addition, as extreme wave form, there is a local non-linear contraction of the wave-group around the crest whichmore » leads to a localised broadening of the wave spectrum which the bandwidth limited non-linear Schrödinger Equations struggle to capture. This limitation occurs for waves of moderate steepness and a narrow underlying spectrum.« less

The picosecond structure of ultra-fast rogue waves

NASA Astrophysics Data System (ADS)

Klein, Avi; Shahal, Shir; Masri, Gilad; Duadi, Hamootal; Sulimani, Kfir; Lib, Ohad; Steinberg, Hadar; Kolpakov, Stanislav A.; Fridman, Moti

2018-02-01

We investigated ultrafast rogue waves in fiber lasers and found three different patterns of rogue waves: single- peaks, twin-peaks, and triple-peaks. The statistics of the different patterns as a function of the pump power of the laser reveals that the probability for all rogue waves patterns increase close to the laser threshold. We developed a numerical model which prove that the ultrafast rogue waves patterns result from both the polarization mode dispersion in the fiber and the non-instantaneous nature of the saturable absorber. This discovery reveals that there are three different types of rogue waves in fiber lasers: slow, fast, and ultrafast, which relate to three different time-scales and are governed by three different sets of equations: the laser rate equations, the nonlinear Schrodinger equation, and the saturable absorber equations, accordingly. This discovery is highly important for analyzing rogue waves and other extreme events in fiber lasers and can lead to realizing types of rogue waves which were not possible so far such as triangular rogue waves.

Reduction of the equation for lower hybrid waves in a plasma to a nonlinear Schroedinger equation

NASA Technical Reports Server (NTRS)

Karney, C. F. F.

1977-01-01

Equations describing the nonlinear propagation of waves in an anisotropic plasma are rarely exactly soluble. However it is often possible to make approximations that reduce the exact equations into a simpler equation. The use of MACSYMA to make such approximations, and so reduce the equation describing lower hybrid waves into the nonlinear Schrodinger equation which is soluble by the inverse scattering method is demonstrated. MACSYMA is used at several stages in the calculation only because there is a natural division between calculations that are easiest done by hand, and those that are easiest done by machine.

Technique for handling wave propagation specific effects in biological tissue: mapping of the photon transport equation to Maxwell's equations.

PubMed

Handapangoda, Chintha C; Premaratne, Malin; Paganin, David M; Hendahewa, Priyantha R D S

2008-10-27

A novel algorithm for mapping the photon transport equation (PTE) to Maxwell's equations is presented. Owing to its accuracy, wave propagation through biological tissue is modeled using the PTE. The mapping of the PTE to Maxwell's equations is required to model wave propagation through foreign structures implanted in biological tissue for sensing and characterization of tissue properties. The PTE solves for only the magnitude of the intensity but Maxwell's equations require the phase information as well. However, it is possible to construct the phase information approximately by solving the transport of intensity equation (TIE) using the full multigrid algorithm.

Asymptotics of QCD traveling waves with fluctuations and running coupling effects

NASA Astrophysics Data System (ADS)

Beuf, Guillaume

2008-09-01

Extending the Balitsky-Kovchegov (BK) equation independently to running coupling or to fluctuation effects due to pomeron loops is known to lead in both cases to qualitative changes of the traveling-wave asymptotic solutions. In this paper we study the extension of the forward BK equation, including both running coupling and fluctuations effects, extending the method developed for the fixed coupling case [E. Brunet, B. Derrida, A.H. Mueller, S. Munier, Phys. Rev. E 73 (2006) 056126, cond-mat/0512021]. We derive the exact asymptotic behavior in rapidity of the probabilistic distribution of the saturation scale.

Traveling wave and soliton solutions of coupled nonlinear Schrödinger equations with harmonic potential and variable coefficients.

PubMed

Zhong, Wei-Ping; Belić, Milivoj

2010-10-01

Exact traveling wave and soliton solutions, including the bright-bright and dark-dark soliton pairs, are found for the system of two coupled nonlinear Schrödinger equations with harmonic potential and variable coefficients, by employing the homogeneous balance principle and the F-expansion technique. A kind of shape-changing soliton collision is identified in the system. The collision is essentially elastic between the two solitons with opposite velocities. Our results demonstrate that the dynamics of solitons can be controlled by selecting the diffraction, nonlinearity, and gain coefficients.

A pseudo energy-invariant method for relativistic wave equations with Riesz space-fractional derivatives

NASA Astrophysics Data System (ADS)

Macías-Díaz, J. E.; Hendy, A. S.; De Staelen, R. H.

2018-03-01

In this work, we investigate a general nonlinear wave equation with Riesz space-fractional derivatives that generalizes various classical hyperbolic models, including the sine-Gordon and the Klein-Gordon equations from relativistic quantum mechanics. A finite-difference discretization of the model is provided using fractional centered differences. The method is a technique that is capable of preserving an energy-like quantity at each iteration. Some computational comparisons against solutions available in the literature are performed in order to assess the capability of the method to preserve the invariant. Our experiments confirm that the technique yields good approximations to the solutions considered. As an application of our scheme, we provide simulations that confirm, for the first time in the literature, the presence of the phenomenon of nonlinear supratransmission in Riesz space-fractional Klein-Gordon equations driven by a harmonic perturbation at the boundary.

Nonlinear waves in electron-positron-ion plasmas including charge separation

NASA Astrophysics Data System (ADS)

Mugemana, A.; Moolla, S.; Lazarus, I. J.

2017-02-01

Nonlinear low-frequency electrostatic waves in a magnetized, three-component plasma consisting of hot electrons, hot positrons and warm ions have been investigated. The electrons and positrons are assumed to have Boltzmann density distributions while the motion of the ions are governed by fluid equations. The system is closed with the Poisson equation. This set of equations is numerically solved for the electric field. The effects of the driving electric field, ion temperature, positron density, ion drift, Mach number and propagation angle are investigated. It is shown that depending on the driving electric field, ion temperature, positron density, ion drift, Mach number and propagation angle, the numerical solutions exhibit waveforms that are sinusoidal, sawtooth and spiky. The introduction of the Poisson equation increased the Mach number required to generate the waveforms but the driving electric field E 0 was reduced. The results are compared with satellite observations.

Kinetic theory for electrostatic waves due to transverse velocity shears

NASA Technical Reports Server (NTRS)

Ganguli, G.; Lee, Y. C.; Palmadesso, P. J.

1988-01-01

A kinetic theory in the form of an integral equation is provided to study the electrostatic oscillations in a collisionless plasma immersed in a uniform magnetic field and a nonuniform transverse electric field. In the low temperature limit the dispersion differential equation is recovered for the transverse Kelvin-Helmholtz modes for arbitrary values of K parallel, where K parallel is the component of the wave vector in the direction of the external magnetic field assumed in the z direction. For higher temperatures the ion-cyclotron-like modes described earlier in the literature by Ganguli, Lee and Plamadesso are recovered. In this article, the integral equation is reduced to a second-order differential equation and a study is made of the kinetic Kelvin-Helmholtz and ion-cyclotron-like modes that constitute the two branches of oscillation in a magnetized plasma including a transverse inhomogeneous dc electric field.

The Dynamics and Evolution of Poles and Rogue Waves for Nonlinear Schrödinger Equations*

NASA Astrophysics Data System (ADS)

Chiu, Tin Lok; Liu, Tian Yang; Chan, Hiu Ning; Wing Chow, Kwok

2017-09-01

Rogue waves are unexpectedly large deviations from equilibrium or otherwise calm positions in physical systems, e.g. hydrodynamic waves and optical beam intensities. The profiles and points of maximum displacements of these rogue waves are correlated with the movement of poles of the exact solutions extended to the complex plane through analytic continuation. Such links are shown to be surprisingly precise for the first order rogue wave of the nonlinear Schrödinger (NLS) and the derivative NLS equations. A computational study on the second order rogue waves of the NLS equation also displays remarkable agreements.

Data-driven discovery of partial differential equations.

PubMed

Rudy, Samuel H; Brunton, Steven L; Proctor, Joshua L; Kutz, J Nathan

2017-04-01

We propose a sparse regression method capable of discovering the governing partial differential equation(s) of a given system by time series measurements in the spatial domain. The regression framework relies on sparsity-promoting techniques to select the nonlinear and partial derivative terms of the governing equations that most accurately represent the data, bypassing a combinatorially large search through all possible candidate models. The method balances model complexity and regression accuracy by selecting a parsimonious model via Pareto analysis. Time series measurements can be made in an Eulerian framework, where the sensors are fixed spatially, or in a Lagrangian framework, where the sensors move with the dynamics. The method is computationally efficient, robust, and demonstrated to work on a variety of canonical problems spanning a number of scientific domains including Navier-Stokes, the quantum harmonic oscillator, and the diffusion equation. Moreover, the method is capable of disambiguating between potentially nonunique dynamical terms by using multiple time series taken with different initial data. Thus, for a traveling wave, the method can distinguish between a linear wave equation and the Korteweg-de Vries equation, for instance. The method provides a promising new technique for discovering governing equations and physical laws in parameterized spatiotemporal systems, where first-principles derivations are intractable.

Analytic solutions for Long's equation and its generalization

NASA Astrophysics Data System (ADS)

Humi, Mayer

2017-12-01

Two-dimensional, steady-state, stratified, isothermal atmospheric flow over topography is governed by Long's equation. Numerical solutions of this equation were derived and used by several authors. In particular, these solutions were applied extensively to analyze the experimental observations of gravity waves. In the first part of this paper we derive an extension of this equation to non-isothermal flows. Then we devise a transformation that simplifies this equation. We show that this simplified equation admits solitonic-type solutions in addition to regular gravity waves. These new analytical solutions provide new insights into the propagation and amplitude of gravity waves over topography.

Magnetospheric Whistler Mode Raytracing with the Inclusion of Finite Electron and ion Temperature

NASA Astrophysics Data System (ADS)

Maxworth, Ashanthi S.

Whistler mode waves are a type of a low frequency (100 Hz - 30 kHz) wave, which exists only in a magnetized plasma. These waves play a major role in Earth's magnetosphere. Due to the impact of whistler mode waves in many fields such as space weather, satellite communications and lifetime of space electronics, it is important to accurately predict the propagation path of these waves. The method used to determine the propagation path of whistler waves is called numerical raytracing. Numerical raytracing determines the power flow path of the whistler mode waves by solving a set of equations known as the Haselgrove's equations. In the majority of the previous work, raytracing was implemented assuming a cold background plasma (0 K), but the actual magnetosphere is at a temperature of about 1 eV (11600 K). In this work we have modified the numerical raytracing algorithm to work at finite electron and ion temperatures. The finite temperature effects have also been introduced into the formulations for linear cyclotron resonance wave growth and Landau damping, which are the primary mechanisms for whistler mode growth and attenuation in the magnetosphere. Including temperature increases the complexity of numerical raytracing, but the overall effects are mostly limited to increasing the group velocity of the waves at highly oblique wave normal angles.

Numerical solution of the wave equation with variable wave speed on nonconforming domains by high-order difference potentials

NASA Astrophysics Data System (ADS)

Britt, S.; Tsynkov, S.; Turkel, E.

2018-02-01

We solve the wave equation with variable wave speed on nonconforming domains with fourth order accuracy in both space and time. This is accomplished using an implicit finite difference (FD) scheme for the wave equation and solving an elliptic (modified Helmholtz) equation at each time step with fourth order spatial accuracy by the method of difference potentials (MDP). High-order MDP utilizes compact FD schemes on regular structured grids to efficiently solve problems on nonconforming domains while maintaining the design convergence rate of the underlying FD scheme. Asymptotically, the computational complexity of high-order MDP scales the same as that for FD.

Dispersive traveling wave solutions of the Equal-Width and Modified Equal-Width equations via mathematical methods and its applications

NASA Astrophysics Data System (ADS)

Lu, Dianchen; Seadawy, Aly R.; Ali, Asghar

2018-06-01

The Equal-Width and Modified Equal-Width equations are used as a model in partial differential equations for the simulation of one-dimensional wave transmission in nonlinear media with dispersion processes. In this article we have employed extend simple equation method and the exp(-varphi(ξ)) expansion method to construct the exact traveling wave solutions of equal width and modified equal width equations. The obtained results are novel and have numerous applications in current areas of research in mathematical physics. It is exposed that our method, with the help of symbolic computation, provides a effective and powerful mathematical tool for solving different kind nonlinear wave problems.

Explicit solutions from eigenfunction symmetry of the Korteweg-de Vries equation.

PubMed

Hu, Xiao-Rui; Lou, Sen-Yue; Chen, Yong

2012-05-01

In nonlinear science, it is very difficult to find exact interaction solutions among solitons and other kinds of complicated waves such as cnoidal waves and Painlevé waves. Actually, even if for the most well-known prototypical models such as the Kortewet-de Vries (KdV) equation and the Kadomtsev-Petviashvili (KP) equation, this kind of problem has not yet been solved. In this paper, the explicit analytic interaction solutions between solitary waves and cnoidal waves are obtained through the localization procedure of nonlocal symmetries which are related to Darboux transformation for the well-known KdV equation. The same approach also yields some other types of interaction solutions among different types of solutions such as solitary waves, rational solutions, Bessel function solutions, and/or general Painlevé II solutions.

Time dependent wave envelope finite difference analysis of sound propagation

NASA Technical Reports Server (NTRS)

Baumeister, K. J.

1984-01-01

A transient finite difference wave envelope formulation is presented for sound propagation, without steady flow. Before the finite difference equations are formulated, the governing wave equation is first transformed to a form whose solution tends not to oscillate along the propagation direction. This transformation reduces the required number of grid points by an order of magnitude. Physically, the transformed pressure represents the amplitude of the conventional sound wave. The derivation for the wave envelope transient wave equation and appropriate boundary conditions are presented as well as the difference equations and stability requirements. To illustrate the method, example solutions are presented for sound propagation in a straight hard wall duct and in a two dimensional straight soft wall duct. The numerical results are in good agreement with exact analytical results.

Fully three-dimensional direct numerical simulation of a plunging breaker

NASA Astrophysics Data System (ADS)

Lubin, Pierre; Vincent, Stéphane; Caltagirone, Jean-Paul; Abadie, Stéphane

2003-07-01

The scope of this paper is to show the results obtained for simulating three-dimensional breaking waves by solving the Navier-Stokes equations in air and water. The interface tracking is achieved by a Lax-Wendroff TVD scheme (Total Variation Diminishing), which is able to handle interface reconnections. We first present the equations and the numerical methods used in this work. We then proceed to the study of a three-dimensional plunging breaking wave, using initial conditions corresponding to unstable periodic sinusoidal waves of large amplitudes. We compare the results obtained for two simulations, a longshore depth perturbation has been introduced in the solution of the flow equations in order to see the transition from a two-dimensional velocity field to a fully three-dimensional one after plunging. Breaking processes including overturning, splash-up and breaking induced vortex-like motion beneath the surface are presented and discussed. To cite this article: P. Lubin et al., C. R. Mecanique 331 (2003).

Seismic modeling with radial basis function-generated finite differences (RBF-FD) – a simplified treatment of interfaces

DOE Office of Scientific and Technical Information (OSTI.GOV)

Martin, Bradley, E-mail: brma7253@colorado.edu; Fornberg, Bengt, E-mail: Fornberg@colorado.edu

In a previous study of seismic modeling with radial basis function-generated finite differences (RBF-FD), we outlined a numerical method for solving 2-D wave equations in domains with material interfaces between different regions. The method was applicable on a mesh-free set of data nodes. It included all information about interfaces within the weights of the stencils (allowing the use of traditional time integrators), and was shown to solve problems of the 2-D elastic wave equation to 3rd-order accuracy. In the present paper, we discuss a refinement of that method that makes it simpler to implement. It can also improve accuracy formore » the case of smoothly-variable model parameter values near interfaces. We give several test cases that demonstrate the method solving 2-D elastic wave equation problems to 4th-order accuracy, even in the presence of smoothly-curved interfaces with jump discontinuities in the model parameters.« less

Bound states and propagating modes in quantum wires with sharp bends and/or constrictions

NASA Astrophysics Data System (ADS)

Razavy, M.

1997-06-01

A number of interesting problems of quantum wires with different geometries can be studied with the help of conformal mapping. These include crossed wires, twisting wires, conductors with constrictions, and wires with a bend. Here the Helmholz equation with Dirichlet boundary condition on the surface of the wire is transformed to a Schröautdinger-like equation with an energy-dependent nonseparable potential but with boundary conditions given on two straight lines. By expanding the wave function in terms of the Fourier series of one of the variables one obtains an infinite set of coupled ordinary differential equations. Only the propagating modes plus a few of the localized modes contribute significantly to the total wave function. Once the problem is solved, one can express the results in terms of the original variables using the inverse conformal mapping. As an example, the total wave function, the components of the current density, and the bound-state energy for a Γ-shaped quantum wire is calculated in detail.

Radiative Amplification of Acoustic Waves in Hot Stars

NASA Technical Reports Server (NTRS)

Wolf, B. E.

1985-01-01

The discovery of broad P Cygni profiles in early type stars and the detection of X-rays emitted from the envelopes of these stars made it clear, that a considerable amount of mechanical energy has to be present in massive stars. An attack on the problem, which has proven successful when applied to late type stars is proposed. It is possible that acoustic waves form out of random fluctuations, amplify by absorbing momentum from stellar radiation field, steepen into shock waves and dissipate. A stellar atmosphere was constructed, and sinusoidal small amplitude perturbations of specified Mach number and period at the inner boundary was introduced. The partial differential equations of hydrodynamics and the equations of radiation transfer for grey matter were solved numerically. The equation of motion was augmented by a term which describes the absorption of momentum from the radiation field in the continuum and in lines, including the Doppler effect and allows for the treatment of a large number of lines in the radiative acceleration term.

Seismic modeling with radial basis function-generated finite differences (RBF-FD) - a simplified treatment of interfaces

NASA Astrophysics Data System (ADS)

Martin, Bradley; Fornberg, Bengt

2017-04-01

In a previous study of seismic modeling with radial basis function-generated finite differences (RBF-FD), we outlined a numerical method for solving 2-D wave equations in domains with material interfaces between different regions. The method was applicable on a mesh-free set of data nodes. It included all information about interfaces within the weights of the stencils (allowing the use of traditional time integrators), and was shown to solve problems of the 2-D elastic wave equation to 3rd-order accuracy. In the present paper, we discuss a refinement of that method that makes it simpler to implement. It can also improve accuracy for the case of smoothly-variable model parameter values near interfaces. We give several test cases that demonstrate the method solving 2-D elastic wave equation problems to 4th-order accuracy, even in the presence of smoothly-curved interfaces with jump discontinuities in the model parameters.

A high-order time-parallel scheme for solving wave propagation problems via the direct construction of an approximate time-evolution operator

DOE Office of Scientific and Technical Information (OSTI.GOV)

Haut, T. S.; Babb, T.; Martinsson, P. G.

2015-06-16

Our manuscript demonstrates a technique for efficiently solving the classical wave equation, the shallow water equations, and, more generally, equations of the form ∂u/∂t=Lu∂u/∂t=Lu, where LL is a skew-Hermitian differential operator. The idea is to explicitly construct an approximation to the time-evolution operator exp(τL)exp(τL) for a relatively large time-step ττ. Recently developed techniques for approximating oscillatory scalar functions by rational functions, and accelerated algorithms for computing functions of discretized differential operators are exploited. Principal advantages of the proposed method include: stability even for large time-steps, the possibility to parallelize in time over many characteristic wavelengths and large speed-ups over existingmore » methods in situations where simulation over long times are required. Numerical examples involving the 2D rotating shallow water equations and the 2D wave equation in an inhomogenous medium are presented, and the method is compared to the 4th order Runge–Kutta (RK4) method and to the use of Chebyshev polynomials. The new method achieved high accuracy over long-time intervals, and with speeds that are orders of magnitude faster than both RK4 and the use of Chebyshev polynomials.« less

The stability of freak waves with regard to external impact and perturbation of initial data

NASA Astrophysics Data System (ADS)

Smirnova, Anna; Shamin, Roman

2014-05-01

We investigate solutions of the equations, describing freak waves, in perspective of stability with regard to external impact and perturbation of initial data. The modeling of freak waves is based on numerical solution of equations describing a non-stationary potential flow of the ideal fluid with a free surface. We consider the two-dimensional infinitely deep flow. For waves modeling we use the equations in conformal variables. The variant of these equations is offered in [1]. Mathematical correctness of these equations was discussed in [2]. These works establish the uniqueness of solutions, offer the effective numerical solution calculation methods, prove the numerical convergence of these methods. The important aspect of numerical modeling of freak waves is the stability of solutions, describing these waves. In this work we study the questions of stability with regards to external impact and perturbation of initial data. We showed the stability of freak waves numerical model, corresponding to the external impact. We performed series of computational experiments with various freak wave initial data and random external impact. This impact means the power density on free surface. In each experiment examine two waves: the wave that was formed by external impact and without one. In all the experiments we see the stability of equation`s solutions. The random external impact practically does not change the time of freak wave formation and its form. Later our work progresses to the investigation of solution's stability under perturbations of initial data. We take the initial data that provide a freak wave and get the numerical solution. In common we take the numerical solution of equation with perturbation of initial data. The computing experiments showed that the freak waves equations solutions are stable under perturbations of initial data.So we can make a conclusion that freak waves are stable relatively external perturbation and perturbation of initial data both. 1. Zakharov V.E., Dyachenko A.I., Vasilyev O.A. New method for numerical simulation of a nonstationary potential flow of incompressible fluid with a free surface// Eur. J.~Mech. B Fluids. 2002. V. 21. P. 283-291. 2. R.V. Shamin. Dynamics of an Ideal Liquid with a Free Surface in Conformal Variables // Journal of Mathematical Sciences, Vol. 160, No. 5, 2009. P. 537-678. 3. R.V. Shamin, V.E. Zakharov, A.I. Dyachenko. How probability for freak wave formation can be found // THE EUROPEAN PHYSICAL JOURNAL - SPECIAL TOPICS Volume 185, Number 1, 113-124, DOI: 10.1140/epjst/e2010-01242-y

Stability and instability of periodic travelling wave solutions for the critical Korteweg-de Vries and nonlinear Schrödinger equations

NASA Astrophysics Data System (ADS)

Angulo Pava, Jaime; Natali, Fábio M. Amorin

2009-04-01

In this paper we establish new results about the existence, stability, and instability of periodic travelling wave solutions related to the critical Korteweg-de Vries equation ut+5u4ux+u=0, and the critical nonlinear Schrödinger equation ivt+v+|v=0. The periodic travelling wave solutions obtained in our study tend to the classical solitary wave solutions in the infinite wavelength scenario. The stability approach is based on the theory developed by Angulo & Natali in [J. Angulo, F. Natali, Positivity properties of the Fourier transform and the stability of periodic travelling wave solutions, SIAM J. Math. Anal. 40 (2008) 1123-1151] for positive periodic travelling wave solutions associated to dispersive evolution equations of Korteweg-de Vries type. The instability approach is based on an extension to the periodic setting of arguments found in Bona & Souganidis & Strauss [J.L. Bona, P.E. Souganidis, W.A. Strauss, Stability and instability of solitary waves of Korteweg-de Vries type, Proc. Roy. Soc. London Ser. A 411 (1987) 395-412]. Regarding the critical Schrödinger equation stability/instability theories similar to the critical Korteweg-de Vries equation are obtained by using the classical Grillakis & Shatah & Strauss theory in [M. Grillakis, J. Shatah, W. Strauss, Stability theory of solitary waves in the presence of symmetry II, J. Funct. Anal. 94 (1990) 308-348; M. Grillakis, J. Shatah, W. Strauss, Stability theory of solitary waves in the presence of symmetry I, J. Funct. Anal. 74 (1987) 160-197]. The arguments presented in this investigation have prospects for the study of the stability of periodic travelling wave solutions of other nonlinear evolution equations.

A highly accurate finite-difference method with minimum dispersion error for solving the Helmholtz equation

NASA Astrophysics Data System (ADS)

Wu, Zedong; Alkhalifah, Tariq

2018-07-01

Numerical simulation of the acoustic wave equation in either isotropic or anisotropic media is crucial to seismic modeling, imaging and inversion. Actually, it represents the core computation cost of these highly advanced seismic processing methods. However, the conventional finite-difference method suffers from severe numerical dispersion errors and S-wave artifacts when solving the acoustic wave equation for anisotropic media. We propose a method to obtain the finite-difference coefficients by comparing its numerical dispersion with the exact form. We find the optimal finite difference coefficients that share the dispersion characteristics of the exact equation with minimal dispersion error. The method is extended to solve the acoustic wave equation in transversely isotropic (TI) media without S-wave artifacts. Numerical examples show that the method is highly accurate and efficient.

Mass, Energy, Space And Time Systemic Theory---MEST

NASA Astrophysics Data System (ADS)

Cao, Dayong

2010-03-01

Things have their physical system of the mass, energy, space and time of themselves-MEST. The matter have the physical systemic moel like that the mass-energy is center and the space-time is around. The time is from the frequency of wave, the space is from the amplitude of wave. What is the physical effection of the wave. The gravity and inertial force is from the wave. Not only the planets have the mass and the kinetic energy, but also it have the wave and the wave energy. According to the equivalence principle of the general relativity, there is the equation: ma=mg and mv^2 /2= δmc^2. The energy equation of the planets: E=mv^2=mgr (v is velocity) be bring put forward. In quantum mechanics, according to the quantum light theory and the de Broglie's theory , there are the equation of the wave: E=hν, p=h/λ (h is Planck constant, p is momentum, λ is the wavelengh), and there is the equation of the wave: E=mc^2. So the energy equation of the planets: E=mv^2 = mv^2 /2 + δmc^2 (mv^2 /2= δmc^2 ) be bring put forward. The equation: δmc^2 show that the planets have the wave of itself, and the wave give the planets the energy. So it do not fall from the heaven. When the matter go into the heaven, it need get the wave energy (like the potential energy). So we can make a new light-flight with the light-driving force.

Dispersive solitary wave solutions of Kadomtsev-Petviashvili and modified Kadomtsev-Petviashvili dynamical equations in unmagnetized dust plasma

NASA Astrophysics Data System (ADS)

Seadawy, A. R.; El-Rashidy, K.

2018-03-01

The Kadomtsev-Petviashvili (KP) and modified KP equations are two of the most universal models in nonlinear wave theory, which arises as a reduction of system with quadratic nonlinearity which admit weakly dispersive waves. The generalized extended tanh method and the F-expansion method are used to derive exact solitary waves solutions of KP and modified KP equations. The region of solutions are displayed graphically.

Exact finite difference schemes for the non-linear unidirectional wave equation

NASA Technical Reports Server (NTRS)

Mickens, R. E.

1985-01-01

Attention is given to the construction of exact finite difference schemes for the nonlinear unidirectional wave equation that describes the nonlinear propagation of a wave motion in the positive x-direction. The schemes constructed for these equations are compared with those obtained by using the usual procedures of numerical analysis. It is noted that the order of the exact finite difference models is equal to the order of the differential equation.

An algorithm for solving the perturbed gas dynamic equations

NASA Technical Reports Server (NTRS)

Davis, Sanford

1993-01-01

The present application of a compact, higher-order central-difference approximation to the linearized Euler equations illustrates the multimodal character of these equations by means of computations for acoustic, vortical, and entropy waves. Such dissipationless central-difference methods are shown to propagate waves exhibiting excellent phase and amplitude resolution on the basis of relatively large time-steps; they can be applied to wave problems governed by systems of first-order partial differential equations.

Investigation of Solitary wave solutions for Vakhnenko-Parkes equation via exp-function and Exp(-ϕ(ξ))-expansion method.

PubMed

Roshid, Harun-Or; Kabir, Md Rashed; Bhowmik, Rajandra Chadra; Datta, Bimal Kumar

2014-01-01

In this paper, we have described two dreadfully important methods to solve nonlinear partial differential equations which are known as exp-function and the exp(-ϕ(ξ)) -expansion method. Recently, there are several methods to use for finding analytical solutions of the nonlinear partial differential equations. The methods are diverse and useful for solving the nonlinear evolution equations. With the help of these methods, we are investigated the exact travelling wave solutions of the Vakhnenko- Parkes equation. The obtaining soliton solutions of this equation are described many physical phenomena for weakly nonlinear surface and internal waves in a rotating ocean. Further, three-dimensional plots of the solutions such as solitons, singular solitons, bell type solitary wave i.e. non-topological solitons solutions and periodic solutions are also given to visualize the dynamics of the equation.

Generalization of the Euler-type solution to the wave equation

NASA Astrophysics Data System (ADS)

Borisov, Victor V.

2001-08-01

Generalization of the Euler-type solution to the wave equation is given. Peculiarities of the space-time structure of obtained waves are considered. For some particular cases interpretation of these waves as `subliminal' and `superluminal' is discussed. The possibility of description of electromagnetic waves by means of the scalar solutions is shown.

On the integrable elliptic cylindrical Kadomtsev-Petviashvili equation.

PubMed

Khusnutdinova, K R; Klein, C; Matveev, V B; Smirnov, A O

2013-03-01

There exist two versions of the Kadomtsev-Petviashvili (KP) equation, related to the Cartesian and cylindrical geometries of the waves. In this paper, we derive and study a new version, related to the elliptic cylindrical geometry. The derivation is given in the context of surface waves, but the derived equation is a universal integrable model applicable to generic weakly nonlinear weakly dispersive waves. We also show that there exist nontrivial transformations between all three versions of the KP equation associated with the physical problem formulation, and use them to obtain new classes of approximate solutions for water waves.

Two dimensional cylindrical fast magnetoacoustic solitary waves in a dust plasma

DOE Office of Scientific and Technical Information (OSTI.GOV)

Liu Haifeng; Wang Shiqing; Engineering and Technical College of Chengdu University of Technology, Leshan 614000

2011-04-15

The nonlinear fast magnetoacoustic solitary waves in a dust plasma with the combined effects of bounded cylindrical geometry and transverse perturbation are investigated in a new equation. In this regard, cylindrical Kadomtsev-Petviashvili (CKP) equation is derived using the small amplitude perturbation expansion method. Under a suitable coordinate transformation, the CKP equation can be solved analytically. It is shown that the dust cylindrical fast magnetoacoustic solitary waves can exist in the CKP equation. The present investigation may have relevance in the study of nonlinear electromagnetic soliton waves both in laboratory and astrophysical plasmas.

Dispersion relations with crossing symmetry for {pi}{pi}D- and F1-wave amplitudes

DOE Office of Scientific and Technical Information (OSTI.GOV)

Kaminski, R.

Results of implementation of dispersion relations with imposed crossing symmetry condition to description of {pi}{pi}D and F1 wave amplitudes are presented. We use relations with only one subtraction what leads to small uncertainties of results and to strong constraints for tested {pi}{pi} amplitudes. Presented equations are similar to those with one subtraction (so called GKPY equations) and to those with two subtractions (the Roy's equations) for the S and P waves. Numerical calculations are done with the S and P wave input amplitudes tested already with use of the Roy's and GKPY equations.

Propagation of seismic waves in tall buildings

USGS Publications Warehouse

Safak, E.

1998-01-01

A discrete-time wave propagation formulation of the seismic response of tall buildings is introduced. The building is modeled as a layered medium, similar to a layered soil medium, and is subjected to vertically propagating seismic shear waves. Soil layers and the bedrock under the foundation are incorporated in the formulation as additional layers. Seismic response is expressed in terms of the wave travel times between the layers, and the wave reflection and transmission coefficients at the layer interfaces. The equations account for the frequency-dependent filtering effects of the foundation and floor masses. The calculation of seismic response is reduced to a pair of simple finite-difference equations for each layer, which can be solved recursively starting from the bedrock. Compared to the commonly used vibration formulation, the wave propagation formulation provides several advantages, including simplified calculations, better representation of damping, ability to account for the effects of the soil layers under the foundation, and better tools for identification and damage detection from seismic records. Examples presented show the versatility of the method. ?? 1998 John Wiley & Sons, Ltd.

Wave-current interactions in three dimensions: why 3D radiation stresses are not practical

NASA Astrophysics Data System (ADS)

Ardhuin, Fabrice

2017-04-01

The coupling of ocean circulation and wave models is based on a wave-averaged mass and momentum conservation equations. Whereas several equivalent equations for the evolution of the current momentum have been proposed, implemented, and used, the possibility to formulate practical equations for the total momentum, which is the sum of the current and wave momenta, has been obscured by a series of publications. In a recent update on previous derivations, Mellor (J. Phys. Oceanogr. 2015) proposed a new set of wave-forced total momentum equations. Here we show that this derivation misses a term that integrates to zero over the vertical. This is because he went from his depth-integrated eq. (28) to the 3D equation (30) by simply removing the integral, but any extra zero-integrating term can be added. Corrected for this omission, the equations of motion are equivalent to the earlier equations by Mellor (2003) which are correct when expressed in terms of wave-induced pressure, horizontal velocity and vertical displacement. Namely the total momentum evolution is driven by the horizontal divergence of a horizontal momentum flux, ----- --- ∂^s- Sαβ = ^uα^uβ + δαβ ∂ς (^p- g^s) (1) and the vertical divergence of a vertical flux, Sαz = (p^-g^s)∂^s/∂xα, (2) where p is the wave-induced non-hydrostatic pressure, s is the wave-induced vertical displacement, and u^ α is the horizontal wave-induced velocity in direction α. So far, so good. Problems arise when p and s are evaluated. Indeend, Ardhuin et al. (J. Phys. Oceanogr. 2008) showed that, over a sloping bottom ∂Sαβ/∂xβ is of order of the slope, hence a consistent wave forcing requires an estimation of Sαz that must be estimated to first order in the bottom slope. For this, Airy wave theory, i.e. cosh(kz-+-kh) p ≃ ga cosh (kD ) cosψ, (3) is not enough. Ardhuin et al. (2008) has shown that using an exact solution of the Laplace equations the vertical flux can indeed be computed. The alternative of neglecting completely Sαz, as suggested by Mellor (2011) for small slopes, will always generate spurious currents because of the unbalanced forcing ∂Sαβ/∂xβ. Fortunately, there are many explicit versions of the wave-averaged equations without the wave momentum in them (Suzuki and Fox-Kemper 2016), with or without vortex force which are all consistent with the exact 3D equations of Andrews and McIntyre (1978). There is thus no need to stumble again and again on this fundamental problem of vertical momentum flux, which is a flux of wave momentum. The problem simply goes away by writing the equations for the current momentum only, without the problematic wave momentum. The current and wave momentum are coupled by forcing terms, and the wave momentum can be solved in 2D, the vertical distribution of momentum being maintained by the complex flux Sαz.

Modified method of simplest equation: Powerful tool for obtaining exact and approximate traveling-wave solutions of nonlinear PDEs

NASA Astrophysics Data System (ADS)

Vitanov, Nikolay K.

2011-03-01

We discuss the class of equations ∑i,j=0mAij(u){∂iu}/{∂ti}∂+∑k,l=0nBkl(u){∂ku}/{∂xk}∂=C(u) where Aij( u), Bkl( u) and C( u) are functions of u( x, t) as follows: (i) Aij, Bkl and C are polynomials of u; or (ii) Aij, Bkl and C can be reduced to polynomials of u by means of Taylor series for small values of u. For these two cases the above-mentioned class of equations consists of nonlinear PDEs with polynomial nonlinearities. We show that the modified method of simplest equation is powerful tool for obtaining exact traveling-wave solution of this class of equations. The balance equations for the sub-class of traveling-wave solutions of the investigated class of equations are obtained. We illustrate the method by obtaining exact traveling-wave solutions (i) of the Swift-Hohenberg equation and (ii) of the generalized Rayleigh equation for the cases when the extended tanh-equation or the equations of Bernoulli and Riccati are used as simplest equations.

Quantitative Estimation of Seismic Velocity Changes Using Time-Lapse Seismic Data and Elastic-Wave Sensitivity Approach

NASA Astrophysics Data System (ADS)

Denli, H.; Huang, L.

2008-12-01

Quantitative monitoring of reservoir property changes is essential for safe geologic carbon sequestration. Time-lapse seismic surveys have the potential to effectively monitor fluid migration in the reservoir that causes geophysical property changes such as density, and P- and S-wave velocities. We introduce a novel method for quantitative estimation of seismic velocity changes using time-lapse seismic data. The method employs elastic sensitivity wavefields, which are the derivatives of elastic wavefield with respect to density, P- and S-wave velocities of a target region. We derive the elastic sensitivity equations from analytical differentiations of the elastic-wave equations with respect to seismic-wave velocities. The sensitivity equations are coupled with the wave equations in a way that elastic waves arriving in a target reservoir behave as a secondary source to sensitivity fields. We use a staggered-grid finite-difference scheme with perfectly-matched layers absorbing boundary conditions to simultaneously solve the elastic-wave equations and the elastic sensitivity equations. By elastic-wave sensitivities, a linear relationship between relative seismic velocity changes in the reservoir and time-lapse seismic data at receiver locations can be derived, which leads to an over-determined system of equations. We solve this system of equations using a least- square method for each receiver to obtain P- and S-wave velocity changes. We validate the method using both surface and VSP synthetic time-lapse seismic data for a multi-layered model and the elastic Marmousi model. Then we apply it to the time-lapse field VSP data acquired at the Aneth oil field in Utah. A total of 10.5K tons of CO2 was injected into the oil reservoir between the two VSP surveys for enhanced oil recovery. The synthetic and field data studies show that our new method can quantitatively estimate changes in seismic velocities within a reservoir due to CO2 injection/migration.

The numerical solution of the Helmholtz equation for wave propagation problems in underwater acoustics

NASA Technical Reports Server (NTRS)

Bayliss, A.; Goldstein, C. I.; Turkel, E.

1984-01-01

The Helmholtz Equation (-delta-K(2)n(2))u=0 with a variable index of refraction, n, and a suitable radiation condition at infinity serves as a model for a wide variety of wave propagation problems. A numerical algorithm was developed and a computer code implemented that can effectively solve this equation in the intermediate frequency range. The equation is discretized using the finite element method, thus allowing for the modeling of complicated geometrices (including interfaces) and complicated boundary conditions. A global radiation boundary condition is imposed at the far field boundary that is exact for an arbitrary number of propagating modes. The resulting large, non-selfadjoint system of linear equations with indefinite symmetric part is solved using the preconditioned conjugate gradient method applied to the normal equations. A new preconditioner is developed based on the multigrid method. This preconditioner is vectorizable and is extremely effective over a wide range of frequencies provided the number of grid levels is reduced for large frequencies. A heuristic argument is given that indicates the superior convergence properties of this preconditioner.

Numerical Simulations of Self-Focused Pulses Using the Nonlinear Maxwell Equations

NASA Technical Reports Server (NTRS)

Goorjian, Peter M.; Silberberg, Yaron; Kwak, Dochan (Technical Monitor)

1994-01-01

This paper will present results in computational nonlinear optics. An algorithm will be described that solves the full vector nonlinear Maxwell's equations exactly without the approximations that are currently made. Present methods solve a reduced scalar wave equation, namely the nonlinear Schrodinger equation, and neglect the optical carrier. Also, results will be shown of calculations of 2-D electromagnetic nonlinear waves computed by directly integrating in time the nonlinear vector Maxwell's equations. The results will include simulations of 'light bullet' like pulses. Here diffraction and dispersion will be counteracted by nonlinear effects. The time integration efficiently implements linear and nonlinear convolutions for the electric polarization, and can take into account such quantum effects as Kerr and Raman interactions. The present approach is robust and should permit modeling 2-D and 3-D optical soliton propagation, scattering, and switching directly from the full-vector Maxwell's equations. Abstract of a proposed paper for presentation at the meeting NONLINEAR OPTICS: Materials, Fundamentals, and Applications, Hyatt Regency Waikaloa, Waikaloa, Hawaii, July 24-29, 1994, Cosponsored by IEEE/Lasers and Electro-Optics Society and Optical Society of America

Almost analytical Karhunen-Loeve representation of irregular waves based on the prolate spheroidal wave functions

NASA Astrophysics Data System (ADS)

Lee, Gibbeum; Cho, Yeunwoo

2017-11-01

We present an almost analytical new approach to solving the matrix eigenvalue problem or the integral equation in Karhunen-Loeve (K-L) representation of random data such as irregular ocean waves. Instead of solving this matrix eigenvalue problem purely numerically, which may suffer from the computational inaccuracy for big data, first, we consider a pair of integral and differential equations, which are related to the so-called prolate spheroidal wave functions (PSWF). For the PSWF differential equation, the pair of the eigenvectors (PSWF) and eigenvalues can be obtained from a relatively small number of analytical Legendre functions. Then, the eigenvalues in the PSWF integral equation are expressed in terms of functional values of the PSWF and the eigenvalues of the PSWF differential equation. Finally, the analytically expressed PSWFs and the eigenvalues in the PWSF integral equation are used to form the kernel matrix in the K-L integral equation for the representation of exemplary wave data; ordinary irregular waves and rogue waves. We found that the present almost analytical method is better than the conventional data-independent Fourier representation and, also, the conventional direct numerical K-L representation in terms of both accuracy and computational cost. This work was supported by the National Research Foundation of Korea (NRF). (NRF-2017R1D1A1B03028299).

Three-dimensional wave-induced current model equations and radiation stresses

NASA Astrophysics Data System (ADS)

Xia, Hua-yong

2017-08-01

After the approach by Mellor (2003, 2008), the present paper reports on a repeated effort to derive the equations for three-dimensional wave-induced current. Via the vertical momentum equation and a proper coordinate transformation, the phase-averaged wave dynamic pressure is well treated, and a continuous and depth-dependent radiation stress tensor, rather than the controversial delta Dirac function at the surface shown in Mellor (2008), is provided. Besides, a phase-averaged vertical momentum flux over a sloping bottom is introduced. All the inconsistencies in Mellor (2003, 2008), pointed out by Ardhuin et al. (2008) and Bennis and Ardhuin (2011), are overcome in the presently revised equations. In a test case with a sloping sea bed, as shown in Ardhuin et al. (2008), the wave-driving forces derived in the present equations are in good balance, and no spurious vertical circulation occurs outside the surf zone, indicating that Airy's wave theory and the approach of Mellor (2003, 2008) are applicable for the derivation of the wave-induced current model.

Propagation and attenuation of Rayleigh waves in generalized thermoelastic media

NASA Astrophysics Data System (ADS)

Sharma, M. D.

2014-01-01

This study considers the propagation of Rayleigh waves in a generalized thermoelastic half-space with stress-free plane boundary. The boundary has the option of being either isothermal or thermally insulated. In either case, the dispersion equation is obtained in the form of a complex irrational expression due to the presence of radicals. This dispersion equation is rationalized into a polynomial equation, which is solvable, numerically, for exact complex roots. The roots of the dispersion equation are obtained after removing the extraneous zeros of this polynomial equation. Then, these roots are filtered out for the inhomogeneous propagation of waves decaying with depth. Numerical examples are solved to analyze the effects of thermal properties of elastic materials on the dispersion of existing surface waves. For these thermoelastic Rayleigh waves, the behavior of elliptical particle motion is studied inside and at the surface of the medium. Insulation of boundary does play a significant role in changing the speed, amplitude, and polarization of Rayleigh waves in thermoelastic media.

Self-Consistent Model of Magnetospheric Ring Current and Propagating Electromagnetic Ion Cyclotron Waves. 1; Waves in Multi Ion Magnetosphere

NASA Technical Reports Server (NTRS)

Khazanov, G. V.; Gumayunov, K. V.; Gallagher, D. L.; Kozyra, J. U.

2006-01-01

The further development of a self-consistent theoretical model of interacting ring current ions and electromagnetic ion cyclotron waves [Khazanov et al., 2003] is presented. In order to adequately take into account the wave propagation and refraction in a multi-ion plasmasphere, we explicitly include the ray tracing equations in our previous self-consistent model and use the general form of the wave kinetic equation. This is a major new feature of the present model and, to the best of our knowledge, the ray tracing equations for the first time are explicitly employed on a global magnetospheric scale in order to self-consistently simulate spatial, temporal, and spectral evolutions of the ring current and electromagnetic ion cyclotron waves. To demonstrate the effects of EMIC wave propagation and refraction on the EMIC wave energy distributions and evolution we simulate the May 1998 storm. The main findings of our simulation can be summarized as follows. First, due to the density gradient at the plasmapause, the net wave refraction is suppressed, and He(+)-mode grows preferably at plasmapause. This result is in a total agreement with the previous ray tracing studies, and very clear observed in presented B-field spectrograms. Second, comparison the global wave distributions with the results from other ring current model [Kozyra et al., 1997] reveals that our model provides more intense and higher plasmapause organized distributions during the May, 1998 storm period. Finally, the found He(+)-mode energy distributions are not Gaussian distributions, and most important that wave energy can occupy not only the region of generation, i. e. the region of small wave normal angles, but the entire wave normal angle region and even only the region near 90 degrees. The latter is extremely crucial for energy transfer to thermal plasmaspheric electrons by resonant Landau damping, and subsequent downward heat transport and excitation of stable auroral red arcs.

Nonlinear coherent structures of Alfvén wave in a collisional plasma

DOE Office of Scientific and Technical Information (OSTI.GOV)

Jana, Sayanee; Chakrabarti, Nikhil; Ghosh, Samiran

2016-07-15

The Alfvén wave dynamics is investigated in the framework of two-fluid approach in a compressible collisional magnetized plasma. In the finite amplitude limit, the dynamics of the nonlinear Alfvén wave is found to be governed by a modified Korteweg-de Vries Burgers equation (mKdVB). In this mKdVB equation, the electron inertia is found to act as a source of dispersion, and the electron-ion collision serves as a dissipation. The collisional dissipation is eventually responsible for the Burgers term in mKdVB equation. In the long wavelength limit, this weakly nonlinear Alfvén wave is shown to be governed by a damped nonlinear Schrödingermore » equation. Furthermore, these nonlinear equations are analyzed by means of analytical calculation and numerical simulation to elucidate the various aspects of the phase-space dynamics of the nonlinear wave. Results reveal that nonlinear Alfvén wave exhibits the dissipation mediated shock, envelope, and breather like structures. Numerical simulations also predict the formation of dissipative Alfvénic rogue wave, giant breathers, and rogue wave holes. These results are discussed in the context of the space plasma.« less

Prediction of Skin Temperature Distribution in Cosmetic Laser Surgery

NASA Astrophysics Data System (ADS)

Ting, Kuen; Chen, Kuen-Tasnn; Cheng, Shih-Feng; Lin, Wen-Shiung; Chang, Cheng-Ren

2008-01-01

The use of lasers in cosmetic surgery has increased dramatically in the past decade. To achieve minimal damage to tissues, the study of the temperature distribution of skin in laser irradiation is very important. The phenomenon of the thermal wave effect is significant due to the highly focused light energy of lasers in very a short time period. The conventional Pennes equation does not take the thermal wave effect into account, which the thermal relaxation time (τ) is neglected, so it is not sufficient to solve instantaneous heating and cooling problem. The purpose of this study is to solve the thermal wave equation to determine the realistic temperature distribution during laser surgery. The analytic solutions of the thermal wave equation are compared with those of the Pennes equation. Moreover, comparisons are made between the results of the above equations and the results of temperature measurement using an infrared thermal image instrument. The thermal wave equation could likely to predict the skin temperature distribution in cosmetic laser surgery.

Nonlinear Waves In A Stenosed Elastic Tube Filled With Viscous Fluid: Forced Perturbed Korteweg-De Vries Equation

NASA Astrophysics Data System (ADS)

Gaik*, Tay Kim; Demiray, Hilmi; Tiong, Ong Chee

In the present work, treating the artery as a prestressed thin-walled and long circularly cylindrical elastic tube with a mild symmetrical stenosis and the blood as an incompressible Newtonian fluid, we have studied the pro pagation of weakly nonlinear waves in such a composite medium, in the long wave approximation, by use of the reductive perturbation method. By intro ducing a set of stretched coordinates suitable for the boundary value type of problems and expanding the field variables into asymptotic series of the small-ness parameter of nonlinearity and dispersion, we obtained a set of nonlinear differential equations governing the terms at various order. By solving these nonlinear differential equations, we obtained the forced perturbed Korteweg-de Vries equation with variable coefficient as the nonlinear evolution equation. By use of the coordinate transformation, it is shown that this type of nonlinear evolution equation admits a progressive wave solution with variable wave speed.

Optical Kerr spatiotemporal dark extreme waves

NASA Astrophysics Data System (ADS)

Wabnitz, Stefan; Kodama, Yuji; Baronio, Fabio

2018-02-01

We study the existence and propagation of multidimensional dark non-diffractive and non-dispersive spatiotemporal optical wave-packets in nonlinear Kerr media. We report analytically and confirm numerically the properties of spatiotemporal dark lines, X solitary waves and lump solutions of the (2 + 1)D nonlinear Schr odinger equation (NLSE). Dark lines, X waves and lumps represent holes of light on a continuous wave background. These solitary waves are derived by exploiting the connection between the (2 + 1)D NLSE and a well-known equation of hydrodynamics, namely the (2+1)D Kadomtsev-Petviashvili (KP) equation. This finding opens a novel path for the excitation and control of spatiotemporal optical solitary and rogue waves, of hydrodynamic nature.

Nonlinear Drift-Kinetic Equation in the Presence of a Circularly Polarized Wave

NASA Technical Reports Server (NTRS)

Khazanov, G. V.; Krivorutsky, E. N.; Six, N. Frank (Technical Monitor)

2002-01-01

Equations of the single particle motion and nonlinear kinetic equation for plasma in the presence of a circularly polarized wave of arbitrary frequency in the drift approximation are presented. The nonstationarity and inhomogeneity of the plasma-wave system are taken into account. The time dependent part of the ponderomotive force is discussed.

Exact traveling wave solutions for system of nonlinear evolution equations.

PubMed

Khan, Kamruzzaman; Akbar, M Ali; Arnous, Ahmed H

2016-01-01

In this work, recently deduced generalized Kudryashov method is applied to the variant Boussinesq equations, and the (2 + 1)-dimensional breaking soliton equations. As a result a range of qualitative explicit exact traveling wave solutions are deduced for these equations, which motivates us to develop, in the near future, a new approach to obtain unsteady solutions of autonomous nonlinear evolution equations those arise in mathematical physics and engineering fields. It is uncomplicated to extend this method to higher-order nonlinear evolution equations in mathematical physics. And it should be possible to apply the same method to nonlinear evolution equations having more general forms of nonlinearities by utilizing the traveling wave hypothesis.

Shock-wave generation and bubble formation in the retina by lasers

NASA Astrophysics Data System (ADS)

Sun, Jinming; Gerstman, Bernard S.; Li, Bin

2000-06-01

The generation of shock waves and bubbles has been experimentally observed due to absorption of sub-nanosecond laser pulses by melanosomes, which are found in retinal pigment epithelium cells. Both the shock waves and bubbles may be the cause of retinal damage at threshold fluence levels. The theoretical modeling of shock wave parameters such as amplitude, and bubble size, is a complicated problem due to the non-linearity of the phenomena. We have used two different approaches for treating pressure variations in water: the Tait Equation and a full Equation Of State (EOS). The Tait Equation has the advantage of being developed specifically to model pressure variations in water and is therefore simpler, quicker computationally, and allows the liquid to sustain negative pressures. Its disadvantage is that it does not allow for a change of phase, which prevents modeling of bubbles and leads to non-physical behavior such as the sustaining of ridiculously large negative pressures. The full EOS treatment includes more of the true thermodynamic behavior, such as phase changes that produce bubbles and avoids the generation of large negative pressures. Its disadvantage is that the usual stable equilibrium EOS allows for no negative pressures at all, since tensile stress is unstable with respect to a transition to the vapor phase. In addition, the EOS treatment requires longer computational times. In this paper, we compare shock wave generation for various laser pulses using the two different mathematical approaches and determine the laser pulse regime for which the simpler Tait Equation can be used with confidence. We also present results of our full EOS treatment in which both shock waves and bubbles are simultaneously modeled.

Discrete rational and breather solution in the spatial discrete complex modified Korteweg-de Vries equation and continuous counterparts.

PubMed

Zhao, Hai-Qiong; Yu, Guo-Fu

2017-04-01

In this paper, a spatial discrete complex modified Korteweg-de Vries equation is investigated. The Lax pair, conservation laws, Darboux transformations, and breather and rational wave solutions to the semi-discrete system are presented. The distinguished feature of the model is that the discrete rational solution can possess new W-shape rational periodic-solitary waves that were not reported before. In addition, the first-order rogue waves reach peak amplitudes which are at least three times of the background amplitude, whereas their continuous counterparts are exactly three times the constant background. Finally, the integrability of the discrete system, including Lax pair, conservation laws, Darboux transformations, and explicit solutions, yields the counterparts of the continuous system in the continuum limit.

Electromagnetic Ion Cyclotron Wavefields in a Realistic Dipole Field

NASA Astrophysics Data System (ADS)

Denton, R. E.

2018-02-01

The latitudinal distribution and properties of electromagnetic ion cyclotron (EMIC) waves determine the total effect of those waves on relativistic electrons. Here we describe the latitudinal variation of EMIC waves simulated self-consistently in a dipole magnetic field for a plasmasphere or plume-like plasma at geostationary orbit with cold H+, He+, and O+ and hot protons with temperature anisotropy. The waves grow as they propagate away from the magnetic equator to higher latitude, while the wave vector turns outward radially and the polarization becomes linear. We calculate the detailed wave spectrum in four latitudinal ranges varying from magnetic latitude (MLAT) close to 0° (magnetic equator) up to 21°. The strongest waves are propagating away from the magnetic equator, but some wave power propagating toward the magnetic equator is observed due to local generation (especially close to the magnetic equator) or reflection. The He band waves, which are generated relatively high up on their dispersion surface, are able to propagate all the way to MLAT = 21°, but the H band waves experience frequency filtering, with no equatorial waves propagating to MLAT = 21° and only the higher-frequency waves propagating to MLAT = 14°. The result is that the wave power averaged k∥, which determines the relativistic electron minimum resonance energy, scales like the inverse of the local magnetic field for the He mode, whereas it is almost constant for the H mode. While the perpendicular wave vector turns outward, it broadens. These wavefields should be useful for simulations of radiation belt particle dynamics.

New envelope solitons for Gerdjikov-Ivanov model in nonlinear fiber optics

NASA Astrophysics Data System (ADS)

Triki, Houria; Alqahtani, Rubayyi T.; Zhou, Qin; Biswas, Anjan

2017-11-01

Exact soliton solutions in a class of derivative nonlinear Schrödinger equations including a pure quintic nonlinearity are investigated. By means of the coupled amplitude-phase formulation, we derive a nonlinear differential equation describing the evolution of the wave amplitude in the non-Kerr quintic media. The resulting amplitude equation is then solved to get exact analytical chirped bright, kink, antikink, and singular soliton solutions for the model. It is also shown that the nonlinear chirp associated with these solitons is crucially dependent on the wave intensity and related to self-steepening and group velocity dispersion parameters. Parametric conditions on physical parameters for the existence of chirped solitons are also presented. These localized structures exist due to a balance among quintic nonlinearity, group velocity dispersion, and self-steepening effects.

Seismic waves in a self-gravitating planet

NASA Astrophysics Data System (ADS)

Brazda, Katharina; de Hoop, Maarten V.; Hörmann, Günther

2013-04-01

The elastic-gravitational equations describe the propagation of seismic waves including the effect of self-gravitation. We rigorously derive and analyze this system of partial differential equations and boundary conditions for a general, uniformly rotating, elastic, but aspherical, inhomogeneous, and anisotropic, fluid-solid earth model, under minimal assumptions concerning the smoothness of material parameters and geometry. For this purpose we first establish a consistent mathematical formulation of the low regularity planetary model within the framework of nonlinear continuum mechanics. Using calculus of variations in a Sobolev space setting, we then show how the weak form of the linearized elastic-gravitational equations directly arises from Hamilton's principle of stationary action. Finally we prove existence and uniqueness of weak solutions by the method of energy estimates and discuss additional regularity properties.

Relative sideband amplitudes versus modulation index for common functions using frequency and phase modulation. [for design and testing of communication system

NASA Technical Reports Server (NTRS)

Stocklin, F.

1973-01-01

The equations defining the amplitude of sidebands resulting from either frequency modulation or phase modulation by either square wave, sine wave, sawtooth or triangular modulating functions are presented. Spectral photographs and computer generated tables of modulation index vs. relative sideband amplitudes are also included.

Schroedinger's Wave Structure of Matter (WSM)

NASA Astrophysics Data System (ADS)

Wolff, Milo; Haselhurst, Geoff

2009-10-01

The puzzling electron is due to the belief that it is a discrete particle. Einstein deduced this structure was impossible since Nature does not allow the discrete particle. Clifford (1876) rejected discrete matter and suggested structures in `space'. Schroedinger, (1937) also eliminated discrete particles writing: What we observe as material bodies and forces are nothing but shapes and variations in the structure of space. Particles are just schaumkommen (appearances). He rejected wave-particle duality. Schroedinger's concept was developed by Milo Wolff and Geoff Haselhurst (SpaceAndMotion.com) using the Scalar Wave Equation to find spherical wave solutions in a 3D quantum space. This WSM, the origin of all the Natural Laws, contains all the electron's properties including the Schroedinger Equation. The origin of Newton's Law F=ma is no longer a puzzle; It originates from Mach's principle of inertia (1883) that depends on the space medium and the WSM. Carver Mead (1999) at CalTech used the WSM to design Intel micro-chips correcting errors of Maxwell's magnetic Equations. Applications of the WSM also describe matter at molecular dimensions: alloys, catalysts, biology and medicine, molecular computers and memories. See ``Schroedinger's Universe'' - at Amazon.com

The Universe according to Schroedinger and Milo

NASA Astrophysics Data System (ADS)

Wolff, Milo

2009-10-01

The puzzling electron is due to the belief that it is a discrete particle. Schroedinger, (1937) eliminated discrete particles writing: What we observe as material bodies and forces are nothing but shapes and variations in the structure of space. Particles are just schaumkommen (appearances). Thus he rejected wave-particle duality. Schroedinger's concept was developed by Milo Wolff using a Scalar Wave Equation in 3D quantum space to find wave solutions. The resulting Wave Structure of Matter (WSM) contains all the electron's properties including the Schroedinger Equation. Further, Newton's Law F=ma is no longer a puzzle; It originates from Mach's principle of inertia (1883) that depends on the space medium and the WSM. These the origin of all the Natural Laws. Carver Mead (1999) at CalTech used the WSM to design Intel micro-chips and to correct errors of Maxwell's Equations. Applications of the WSM describe matter at molecular dimensions: Industrial alloys, catalysts, biology and medicine, molecular computers and memories. See book ``Schroedinger's Universe'' - at Amazon.com. Pioneers of the WSM are growing rapidly. Some are: SpaceAndMotion.com, QuantumMatter.com, treeincarnation.com/audio/milowolff.htm, daugerresearch.com/orbitals/index.shtml, glafreniere.com/matter.html =A new Universe.

Schroedinger's Wave Structure of Matter (WSM)

NASA Astrophysics Data System (ADS)

Wolff, Milo

2009-05-01

The puzzling electron is due to the belief that it is a discrete particle. Einstein deduced this structure impossible since Nature does not match the discrete particle. Clifford (1876) rejected discrete matter and suggested structures in `space'. Schroedinger, (1937) also eliminated discrete particles writing: What we observe as material bodies and forces are nothing but shapes and variations in the structure of space. Particles are just schaumkommen (appearances). He rejected wave-particle duality. Schroedinger's concept was developed by Milo Wolff and Geoff Haselhurst (http://www.SpaceAndMotion.com) using the Scalar Wave Equation to find spherical wave solutions in a 3D quantum space. This WSM is the origin of all the Natural Laws; thus it contains all the electron's properties including the Schroedinger Equation. The origin of Newton's Law F=ma is no longer a puzzle; it is shown to originate from Mach's principle of inertia (1883) that depends on the space medium. Carver Mead (1999) applied the WSM to design Intel micro-chips correcting errors of Maxwell's magnetic Equations. Applications of the WSM describe matter at molecular dimensions: alloys, catalysts, the mechanisms of biology and medicine, molecular computers and memories. See http://www.amazon.com/Schro at Amazon.com.

Nearshore Wave and Circulation Modelling

DTIC Science & Technology

1998-02-01

1995), "The unified Kadomtsev - Petviashvili equation for interfacial waves," J. Fluid Mech., 288, 383-408. Chen, Y. and Liu, P. L.-F. (1996), "On...modified Kadomtsev - Petviashvili equation for interfacial wave propagation near the critical depth level," Wave Motion (to appear). Cox, D. T. and Kobayashi...94-13. Chen, Y. and Liu, P.L.-F. (1995), "Numerical Study of the Unified Kadomtsev - Petviashvili Equation ," CACR-95-04. Chen, Y. and Liu, P.L.-F

A Relation Between the Eikonal Equation Associated to a Potential Energy Surface and a Hyperbolic Wave Equation.

PubMed

Bofill, Josep Maria; Quapp, Wolfgang; Caballero, Marc

2012-12-11

The potential energy surface (PES) of a molecule can be decomposed into equipotential hypersurfaces. We show in this article that the hypersurfaces are the wave fronts of a certain hyperbolic partial differential equation, a wave equation. It is connected with the gradient lines, or the steepest descent, or the steepest ascent lines of the PES. The energy seen as a reaction coordinate plays the central role in this treatment.

Nonlinear extraordinary wave in dense plasma

DOE Office of Scientific and Technical Information (OSTI.GOV)

Krasovitskiy, V. B., E-mail: krasovit@mail.ru; Turikov, V. A.

2013-10-15

Conditions for the propagation of a slow extraordinary wave in dense magnetized plasma are found. A solution to the set of relativistic hydrodynamic equations and Maxwell’s equations under the plasma resonance conditions, when the phase velocity of the nonlinear wave is equal to the speed of light, is obtained. The deviation of the wave frequency from the resonance frequency is accompanied by nonlinear longitudinal-transverse oscillations. It is shown that, in this case, the solution to the set of self-consistent equations obtained by averaging the initial equations over the period of high-frequency oscillations has the form of an envelope soliton. Themore » possibility of excitation of a nonlinear wave in plasma by an external electromagnetic pulse is confirmed by numerical simulations.« less

Three-dimensional simulation of beam propagation and heat transfer in static gas Cs DPALs using wave optics and fluid dynamics models

NASA Astrophysics Data System (ADS)

Waichman, Karol; Barmashenko, Boris D.; Rosenwaks, Salman

2017-10-01

Analysis of beam propagation, kinetic and fluid dynamic processes in Cs diode pumped alkali lasers (DPALs), using wave optics model and gasdynamic code, is reported. The analysis is based on a three-dimensional, time-dependent computational fluid dynamics (3D CFD) model. The Navier-Stokes equations for momentum, heat and mass transfer are solved by a commercial Ansys FLUENT solver based on the finite volume discretization technique. The CFD code which solves the gas conservation equations includes effects of natural convection and temperature diffusion of the species in the DPAL mixture. The DPAL kinetic processes in the Cs/He/C2H6 gas mixture dealt with in this paper involve the three lowest energy levels of Cs, (1) 62S1/2, (2) 62P1/2 and (3) 62P3/2. The kinetic processes include absorption due to the 1->3 D2 transition followed by relaxation the 3 to 2 fine structure levels and stimulated emission due to the 2->1 D1 transition. Collisional quenching of levels 2 and 3 and spontaneous emission from these levels are also considered. The gas flow conservation equations are coupled to fast-Fourier-transform algorithm for transverse mode propagation to obtain a solution of the scalar paraxial propagation equation for the laser beam. The wave propagation equation is solved by the split-step beam propagation method where the gain and refractive index in the DPAL medium affect the wave amplitude and phase. Using the CFD and beam propagation models, the gas flow pattern and spatial distributions of the pump and laser intensities in the resonator were calculated for end-pumped Cs DPAL. The laser power, DPAL medium temperature and the laser beam quality were calculated as a function of pump power. The results of the theoretical model for laser power were compared to experimental results of Cs DPAL.

Study on monostable and bistable reaction-diffusion equations by iteration of travelling wave maps

NASA Astrophysics Data System (ADS)

Yi, Taishan; Chen, Yuming

2017-12-01

In this paper, based on the iterative properties of travelling wave maps, we develop a new method to obtain spreading speeds and asymptotic propagation for monostable and bistable reaction-diffusion equations. Precisely, for Dirichlet problems of monostable reaction-diffusion equations on the half line, by making links between travelling wave maps and integral operators associated with the Dirichlet diffusion kernel (the latter is NOT invariant under translation), we obtain some iteration properties of the Dirichlet diffusion and some a priori estimates on nontrivial solutions of Dirichlet problems under travelling wave transformation. We then provide the asymptotic behavior of nontrivial solutions in the space-time region for Dirichlet problems. These enable us to develop a unified method to obtain results on heterogeneous steady states, travelling waves, spreading speeds, and asymptotic spreading behavior for Dirichlet problem of monostable reaction-diffusion equations on R+ as well as of monostable/bistable reaction-diffusion equations on R.

Helical localized wave solutions of the scalar wave equation.

PubMed

Overfelt, P L

2001-08-01

A right-handed helical nonorthogonal coordinate system is used to determine helical localized wave solutions of the homogeneous scalar wave equation. Introducing the characteristic variables in the helical system, i.e., u = zeta - ct and v = zeta + ct, where zeta is the coordinate along the helical axis, we can use the bidirectional traveling plane wave representation and obtain sets of elementary bidirectional helical solutions to the wave equation. Not only are these sets bidirectional, i.e., based on a product of plane waves, but they may also be broken up into right-handed and left-handed solutions. The elementary helical solutions may in turn be used to create general superpositions, both Fourier and bidirectional, from which new solutions to the wave equation may be synthesized. These new solutions, based on the helical bidirectional superposition, are members of the class of localized waves. Examples of these new solutions are a helical fundamental Gaussian focus wave mode, a helical Bessel-Gauss pulse, and a helical acoustic directed energy pulse train. Some of these solutions have the interesting feature that their shape and localization properties depend not only on the wave number governing propagation along the longitudinal axis but also on the normalized helical pitch.

Target-in-the-loop beam control: basic considerations for analysis and wave-front sensing

NASA Astrophysics Data System (ADS)

Vorontsov, Mikhail A.; Kolosov, Valeriy

2005-01-01

Target-in-the-loop (TIL) wave propagation geometry represents perhaps the most challenging case for adaptive optics applications that are related to maximization of irradiance power density on extended remotely located surfaces in the presence of dynamically changing refractive-index inhomogeneities in the propagation medium. We introduce a TIL propagation model that uses a combination of the parabolic equation describing coherent outgoing-wave propagation, and the equation describing evolution of the mutual correlation function (MCF) for the backscattered wave (return wave). The resulting evolution equation for the MCF is further simplified by use of the smooth-refractive-index approximation. This approximation permits derivation of the transport equation for the return-wave brightness function, analyzed here by the method of characteristics (brightness function trajectories). The equations for the brightness function trajectories (ray equations) can be efficiently integrated numerically. We also consider wave-front sensors that perform sensing of speckle-averaged characteristics of the wave-front phase (TIL sensors). Analysis of the wave-front phase reconstructed from Shack-Hartmann TIL sensor measurements shows that an extended target introduces a phase modulation (target-induced phase) that cannot be easily separated from the atmospheric-turbulence-related phase aberrations. We also show that wave-front sensing results depend on the extended target shape, surface roughness, and outgoing-beam intensity distribution on the target surface. For targets with smooth surfaces and nonflat shapes, the target-induced phase can contain aberrations. The presence of target-induced aberrations in the conjugated phase may result in a deterioration of adaptive system performance.

Target-in-the-loop beam control: basic considerations for analysis and wave-front sensing.

PubMed

Vorontsov, Mikhail A; Kolosov, Valeriy

2005-01-01

Target-in-the-loop (TIL) wave propagation geometry represents perhaps the most challenging case for adaptive optics applications that are related to maximization of irradiance power density on extended remotely located surfaces in the presence of dynamically changing refractive-index inhomogeneities in the propagation medium. We introduce a TIL propagation model that uses a combination of the parabolic equation describing coherent outgoing-wave propagation, and the equation describing evolution of the mutual correlation function (MCF) for the backscattered wave (return wave). The resulting evolution equation for the MCF is further simplified by use of the smooth-refractive-index approximation. This approximation permits derivation of the transport equation for the return-wave brightness function, analyzed here by the method of characteristics (brightness function trajectories). The equations for the brightness function trajectories (ray equations) can be efficiently integrated numerically. We also consider wave-front sensors that perform sensing of speckle-averaged characteristics of the wave-front phase (TIL sensors). Analysis of the wave-front phase reconstructed from Shack-Hartmann TIL sensor measurements shows that an extended target introduces a phase modulation (target-induced phase) that cannot be easily separated from the atmospheric-turbulence-related phase aberrations. We also show that wave-front sensing results depend on the extended target shape, surface roughness, and outgoing-beam intensity distribution on the target surface. For targets with smooth surfaces and nonflat shapes, the target-induced phase can contain aberrations. The presence of target-induced aberrations in the conjugated phase may result in a deterioration of adaptive system performance.

Assessment of numerical methods for the solution of fluid dynamics equations for nonlinear resonance systems

NASA Technical Reports Server (NTRS)

Przekwas, A. J.; Yang, H. Q.

1989-01-01

The capability of accurate nonlinear flow analysis of resonance systems is essential in many problems, including combustion instability. Classical numerical schemes are either too diffusive or too dispersive especially for transient problems. In the last few years, significant progress has been made in the numerical methods for flows with shocks. The objective was to assess advanced shock capturing schemes on transient flows. Several numerical schemes were tested including TVD, MUSCL, ENO, FCT, and Riemann Solver Godunov type schemes. A systematic assessment was performed on scalar transport, Burgers' and gas dynamic problems. Several shock capturing schemes are compared on fast transient resonant pipe flow problems. A system of 1-D nonlinear hyperbolic gas dynamics equations is solved to predict propagation of finite amplitude waves, the wave steepening, formation, propagation, and reflection of shocks for several hundred wave cycles. It is shown that high accuracy schemes can be used for direct, exact nonlinear analysis of combustion instability problems, preserving high harmonic energy content for long periods of time.

Nonlinear Waves.

DTIC Science & Technology

1988-02-01

in Multi- dimensions II, P.M. Santini and A.S. Fokas, preprint INS#67, 1986. The Recursion Operator of the Kadomtsev - Petviashvili Equation and the...solitons, multidimensional inverse problems, Painleve equations , direct linearizations of certain nonlinear wave equations , DBAR problems, Riemann...the Navy is (a) the recent discovery that many of the equations describing ship hydrodynamics in channels of finite depth obey nonlinear equations

Characteristics of solitary waves in a relativistic degenerate ion beam driven magneto plasma

NASA Astrophysics Data System (ADS)

Deka, Manoj Kr.; Dev, Apul N.; Misra, Amar P.; Adhikary, Nirab C.

2018-01-01

The nonlinear propagation of a small amplitude ion acoustic solitary wave in a relativistic degenerate magneto plasma in the presence of an ion beam is investigated in detail. The nonlinear equations describing the evolution of a solitary wave in the presence of relativistic non-degenerate magnetized positive ions and ion beams including magnetized degenerate relativistic electrons are derived in terms of Zakharov-Kuznetsov (Z-K) equation for such plasma systems. The ion beams which are a ubiquitous ingredient in such plasma systems are found to have a decisive role in the propagation of a solitary wave in such a highly dense plasma system. The conditions of a wave, propagating with typical solitonic characteristics, are examined and discussed in detail under suitable conditions of different physical parameters. Both a subsonic and supersonic wave can propagate in such plasmas bearing different characteristics under different physical situations. A detailed analysis of waves propagating in subsonic and/or supersonic regime is carried out. The ion beam concentrations, magnetic field, as well as ion beam streaming velocity are found to play a momentous role on the control of the amplitude and width of small amplitude perturbation in both weakly (or non-relativistic) and relativistic plasmas.

Model Parameterization and P-wave AVA Direct Inversion for Young's Impedance

NASA Astrophysics Data System (ADS)

Zong, Zhaoyun; Yin, Xingyao

2017-05-01

AVA inversion is an important tool for elastic parameters estimation to guide the lithology prediction and "sweet spot" identification of hydrocarbon reservoirs. The product of the Young's modulus and density (named as Young's impedance in this study) is known as an effective lithology and brittleness indicator of unconventional hydrocarbon reservoirs. Density is difficult to predict from seismic data, which renders the estimation of the Young's impedance inaccurate in conventional approaches. In this study, a pragmatic seismic AVA inversion approach with only P-wave pre-stack seismic data is proposed to estimate the Young's impedance to avoid the uncertainty brought by density. First, based on the linearized P-wave approximate reflectivity equation in terms of P-wave and S-wave moduli, the P-wave approximate reflectivity equation in terms of the Young's impedance is derived according to the relationship between P-wave modulus, S-wave modulus, Young's modulus and Poisson ratio. This equation is further compared to the exact Zoeppritz equation and the linearized P-wave approximate reflectivity equation in terms of P- and S-wave velocities and density, which illustrates that this equation is accurate enough to be used for AVA inversion when the incident angle is within the critical angle. Parameter sensitivity analysis illustrates that the high correlation between the Young's impedance and density render the estimation of the Young's impedance difficult. Therefore, a de-correlation scheme is used in the pragmatic AVA inversion with Bayesian inference to estimate Young's impedance only with pre-stack P-wave seismic data. Synthetic examples demonstrate that the proposed approach is able to predict the Young's impedance stably even with moderate noise and the field data examples verify the effectiveness of the proposed approach in Young's impedance estimation and "sweet spots" evaluation.

An analytical model of a curved beam with a T shaped cross section

NASA Astrophysics Data System (ADS)

Hull, Andrew J.; Perez, Daniel; Cox, Donald L.

2018-03-01

This paper derives a comprehensive analytical dynamic model of a closed circular beam that has a T shaped cross section. The new model includes in-plane and out-of-plane vibrations derived using continuous media expressions which produces results that have a valid frequency range above those available from traditional lumped parameter models. The web is modeled using two-dimensional elasticity equations for in-plane motion and the classical flexural plate equation for out-of-plane motion. The flange is modeled using two sets of Donnell shell equations: one for the left side of the flange and one for the right side of the flange. The governing differential equations are solved with unknown wave propagation coefficients multiplied by spatial domain and time domain functions which are inserted into equilibrium and continuity equations at the intersection of the web and flange and into boundary conditions at the edges of the system resulting in 24 algebraic equations. These equations are solved to yield the wave propagation coefficients and this produces a solution to the displacement field in all three dimensions. An example problem is formulated and compared to results from finite element analysis.

Consistent three-equation model for thin films

NASA Astrophysics Data System (ADS)

Richard, Gael; Gisclon, Marguerite; Ruyer-Quil, Christian; Vila, Jean-Paul

2017-11-01

Numerical simulations of thin films of newtonian fluids down an inclined plane use reduced models for computational cost reasons. These models are usually derived by averaging over the fluid depth the physical equations of fluid mechanics with an asymptotic method in the long-wave limit. Two-equation models are based on the mass conservation equation and either on the momentum balance equation or on the work-energy theorem. We show that there is no two-equation model that is both consistent and theoretically coherent and that a third variable and a three-equation model are required to solve all theoretical contradictions. The linear and nonlinear properties of two and three-equation models are tested on various practical problems. We present a new consistent three-equation model with a simple mathematical structure which allows an easy and reliable numerical resolution. The numerical calculations agree fairly well with experimental measurements or with direct numerical resolutions for neutral stability curves, speed of kinematic waves and of solitary waves and depth profiles of wavy films. The model can also predict the flow reversal at the first capillary trough ahead of the main wave hump.

Linear Water Waves

NASA Astrophysics Data System (ADS)

Kuznetsov, N.; Maz'ya, V.; Vainberg, B.

2002-08-01

This book gives a self-contained and up-to-date account of mathematical results in the linear theory of water waves. The study of waves has many applications, including the prediction of behavior of floating bodies (ships, submarines, tension-leg platforms etc.), the calculation of wave-making resistance in naval architecture, and the description of wave patterns over bottom topography in geophysical hydrodynamics. The first section deals with time-harmonic waves. Three linear boundary value problems serve as the approximate mathematical models for these types of water waves. The next section uses a plethora of mathematical techniques in the investigation of these three problems. The techniques used in the book include integral equations based on Green's functions, various inequalities between the kinetic and potential energy and integral identities which are indispensable for proving the uniqueness theorems. The so-called inverse procedure is applied to constructing examples of non-uniqueness, usually referred to as 'trapped nodes.'

Exact traveling wave solutions of the KP-BBM equation by using the new approach of generalized (G'/G)-expansion method.

PubMed

Alam, Md Nur; Akbar, M Ali

2013-01-01

The new approach of the generalized (G'/G)-expansion method is an effective and powerful mathematical tool in finding exact traveling wave solutions of nonlinear evolution equations (NLEEs) in science, engineering and mathematical physics. In this article, the new approach of the generalized (G'/G)-expansion method is applied to construct traveling wave solutions of the Kadomtsev-Petviashvili-Benjamin-Bona-Mahony (KP-BBM) equation. The solutions are expressed in terms of the hyperbolic functions, the trigonometric functions and the rational functions. By means of this scheme, we found some new traveling wave solutions of the above mentioned equation.

High-frequency sound waves to eliminate a horizon in the mixmaster universe.

NASA Technical Reports Server (NTRS)

Chitre, D. M.

1972-01-01

From the linear wave equation for small-amplitude sound waves in a curved space-time, there is derived a geodesiclike differential equation for sound rays to describe the motion of wave packets. These equations are applied in the generic, nonrotating, homogeneous closed-model universe (the 'mixmaster universe,' Bianchi type IX). As for light rays described by Doroshkevich and Novikov (DN), these sound rays can circumnavigate the universe near the singularity to remove particle horizons only for a small class of these models and in special directions. Although these results parallel those of DN, different Hamiltonian methods are used for treating the Einstein equations.

Unstable solitary-wave solutions of the generalized Benjamin-Bona-Mahony equation

DOE Office of Scientific and Technical Information (OSTI.GOV)

McKinney, W.R.; Restrepo, J.M.; Bona, J.L.

1994-06-01

The evolution of solitary waves of the gBBM equation is investigated computationally. The experiments confirm previously derived theoretical stability estimates and, more importantly, yield insights into their behavior. For example, highly energetic unstable solitary waves when perturbed are shown to evolve into several stable solitary waves.

Stability of transition waves and positive entire solutions of Fisher-KPP equations with time and space dependence

NASA Astrophysics Data System (ADS)

Shen, Wenxian

2017-09-01

This paper is concerned with the stability of transition waves and strictly positive entire solutions of random and nonlocal dispersal evolution equations of Fisher-KPP type with general time and space dependence, including time and space periodic or almost periodic dependence as special cases. We first show the existence, uniqueness, and stability of strictly positive entire solutions of such equations. Next, we show the stability of uniformly continuous transition waves connecting the unique strictly positive entire solution and the trivial solution zero and satisfying certain decay property at the end close to the trivial solution zero (if it exists). The existence of transition waves has been studied in Liang and Zhao (2010 J. Funct. Anal. 259 857-903), Nadin (2009 J. Math. Pures Appl. 92 232-62), Nolen et al (2005 Dyn. PDE 2 1-24), Nolen and Xin (2005 Discrete Contin. Dyn. Syst. 13 1217-34) and Weinberger (2002 J. Math. Biol. 45 511-48) for random dispersal Fisher-KPP equations with time and space periodic dependence, in Nadin and Rossi (2012 J. Math. Pures Appl. 98 633-53), Nadin and Rossi (2015 Anal. PDE 8 1351-77), Nadin and Rossi (2017 Arch. Ration. Mech. Anal. 223 1239-67), Shen (2010 Trans. Am. Math. Soc. 362 5125-68), Shen (2011 J. Dynam. Differ. Equ. 23 1-44), Shen (2011 J. Appl. Anal. Comput. 1 69-93), Tao et al (2014 Nonlinearity 27 2409-16) and Zlatoš (2012 J. Math. Pures Appl. 98 89-102) for random dispersal Fisher-KPP equations with quite general time and/or space dependence, and in Coville et al (2013 Ann. Inst. Henri Poincare 30 179-223), Rawal et al (2015 Discrete Contin. Dyn. Syst. 35 1609-40) and Shen and Zhang (2012 Comm. Appl. Nonlinear Anal. 19 73-101) for nonlocal dispersal Fisher-KPP equations with time and/or space periodic dependence. The stability result established in this paper implies that the transition waves obtained in many of the above mentioned papers are asymptotically stable for well-fitted perturbation. Up to the author’s knowledge, it is the first time that the stability of transition waves of Fisher-KPP equations with general time and space dependence is studied.

The nonlinear Schrödinger equation and the propagation of weakly nonlinear waves in optical fibers and on the water surface

DOE Office of Scientific and Technical Information (OSTI.GOV)

Chabchoub, A., E-mail: achabchoub@swin.edu.au; Kibler, B.; Finot, C.

2015-10-15

The dynamics of waves in weakly nonlinear dispersive media can be described by the nonlinear Schrödinger equation (NLSE). An important feature of the equation is that it can be derived in a number of different physical contexts; therefore, analogies between different fields, such as for example fiber optics, water waves, plasma waves and Bose–Einstein condensates, can be established. Here, we investigate the similarities between wave propagation in optical Kerr media and water waves. In particular, we discuss the modulation instability (MI) in both media. In analogy to the water wave problem, we derive for Kerr-media the Benjamin–Feir index, i.e. amore » nondimensional parameter related to the probability of formation of rogue waves in incoherent wave trains.« less

KP Equation in a Three-Dimensional Unmagnetized Warm Dusty Plasma with Variable Dust Charge

NASA Astrophysics Data System (ADS)

El-Shorbagy, Kh. H.; Mahassen, Hania; El-Bendary, Atef Ahmed

2017-12-01

In this work, we investigate the propagation of three-dimensional nonlinear dust-acoustic and dust-Coulomb waves in an unmagnetized warm dusty plasma consisting of electrons, ions, and charged dust particles. The grain charge fluctuation is incorporated through the current balance equation. Using the perturbation method, a Kadomtsev-Petviashvili (KP) equation is obtained. It has been shown that the charge fluctuation would modify the wave structures, and the waves in such systems are unstable due to high-order long wave perturbations.

Secondary Bifurcation and Change of Type for Three Dimensional Standing Waves in Shallow Water.

DTIC Science & Technology

1986-02-01

field of standing K-P waves. A set of two non-interacting (to first order) solutions of the K-P equation ( Kadomtsev - Petviashvili 1970). The K-P equation ...P equation was first derived by Kadomtsev & Petviashvili (1970) in their study of the stability of solitary waves to transverse perturbations. A...Scientists, Springer-Verlag 6. B.A. Dubrovin (1981), "Theta Functions and Non-linear Equations ", Russian Mat. Surveys, 36, 11-92 7 B.B. Kadomtsev

Soliton and quasi-periodic wave solutions for b-type Kadomtsev-Petviashvili equation

NASA Astrophysics Data System (ADS)

Singh, Manjit; Gupta, R. K.

2017-11-01

In this paper, truncated Laurent expansion is used to obtain the bilinear equation of a nonlinear evolution equation. As an application of Hirota's method, multisoliton solutions are constructed from the bilinear equation. Extending the application of Hirota's method and employing multidimensional Riemann theta function, one and two-periodic wave solutions are also obtained in a straightforward manner. The asymptotic behavior of one and two-periodic wave solutions under small amplitude limits is presented, and their relations with soliton solutions are also demonstrated.

A Numerical Method for Predicting Rayleigh Surface Wave Velocity in Anisotropic Crystals (Postprint)

DTIC Science & Technology

2017-09-05

generalized version of the equations are very difficult to derive, even in symbolic math languages such as Mathematica. As a result, the equations are...formalism, Math . Mech. Solids 9 (1) (2004) 5–15. [8] M. Destrade, The explicit secular equation for surface acoustic waves in monoclinic elastic crystals...Q. J. Mech. Appl. Math . 55 (2) (2002) 297–311. [10] D. Taylor, Surface waves in anisotropic media: the secular equation and its numerical solution

A novel approach for solitary wave solutions of the generalized fractional Zakharov-Kuznetsov equation

NASA Astrophysics Data System (ADS)

Batool, Fiza; Akram, Ghazala

2018-01-01

In this article the solitary wave solutions of generalized fractional Zakharov-Kuznetsov (GZK) equation which appear in the electrical transmission line model are investigated. The (G'/G)-expansion method is used to obtain the solitary solutions of fractional GZK equation via local fractional derivative. Three classes of solutions, hyperbolic, trigonometric and rational wave solutions of the associated equation are characterized with some free parameters. The obtained solutions reveal that the proposed technique is effective and powerful.

Statistics of extreme waves in the framework of one-dimensional Nonlinear Schrodinger Equation

NASA Astrophysics Data System (ADS)

Agafontsev, Dmitry; Zakharov, Vladimir

2013-04-01

We examine the statistics of extreme waves for one-dimensional classical focusing Nonlinear Schrodinger (NLS) equation, iΨt + Ψxx + |Ψ |2Ψ = 0, (1) as well as the influence of the first nonlinear term beyond Eq. (1) - the six-wave interactions - on the statistics of waves in the framework of generalized NLS equation accounting for six-wave interactions, dumping (linear dissipation, two- and three-photon absorption) and pumping terms, We solve these equations numerically in the box with periodically boundary conditions starting from the initial data Ψt=0 = F(x) + ?(x), where F(x) is an exact modulationally unstable solution of Eq. (1) seeded by stochastic noise ?(x) with fixed statistical properties. We examine two types of initial conditions F(x): (a) condensate state F(x) = 1 for Eq. (1)-(2) and (b) cnoidal wave for Eq. (1). The development of modulation instability in Eq. (1)-(2) leads to formation of one-dimensional wave turbulence. In the integrable case the turbulence is called integrable and relaxes to one of infinite possible stationary states. Addition of six-wave interactions term leads to appearance of collapses that eventually are regularized by the dumping terms. The energy lost during regularization of collapses in (2) is restored by the pumping term. In the latter case the system does not demonstrate relaxation-like behavior. We measure evolution of spectra Ik =< |Ψk|2 >, spatial correlation functions and the PDFs for waves amplitudes |Ψ|, concentrating special attention on formation of "fat tails" on the PDFs. For the classical integrable NLS equation (1) with condensate initial condition we observe Rayleigh tails for extremely large waves and a "breathing region" for middle waves with oscillations of the frequency of waves appearance with time, while nonintegrable NLS equation with dumping and pumping terms (2) with the absence of six-wave interactions α = 0 demonstrates perfectly Rayleigh PDFs without any oscillations with time. In case of the cnoidal wave initial condition we observe severely non-Rayleigh PDFs for the classical NLS equation (1) with the regions corresponding to 2-, 3- and so on soliton collisions clearly seen of the PDFs. Addition of six-wave interactions in Eq. (2) for condensate initial condition results in appearance of non-Rayleigh addition to the PDFs that increase with six-wave interaction constant α and disappears with the absence of six-wave interactions α = 0. References: [1] D.S. Agafontsev, V.E. Zakharov, Rogue waves statistics in the framework of one-dimensional Generalized Nonlinear Schrodinger Equation, arXiv:1202.5763v3.

Effect of EMIC Wave Normal Angle Distribution on Relativistic Electron Scattering Based on the Newly Developed Self-consistent RC/EMIC Waves Model by Khazanov et al. [2006

NASA Technical Reports Server (NTRS)

Khazanov, G. V.; Gallagher, D. L.; Gamayunov, K.

2007-01-01

It is well known that the effects of EMIC waves on RC ion and RB electron dynamics strongly depend on such particle/wave characteristics as the phase-space distribution function, frequency, wave-normal angle, wave energy, and the form of wave spectral energy density. Therefore, realistic characteristics of EMIC waves should be properly determined by modeling the RC-EMIC waves evolution self-consistently. Such a selfconsistent model progressively has been developing by Khaznnov et al. [2002-2006]. It solves a system of two coupled kinetic equations: one equation describes the RC ion dynamics and another equation describes the energy density evolution of EMIC waves. Using this model, we present the effectiveness of relativistic electron scattering and compare our results with previous work in this area of research.

Rogue-wave bullets in a composite (2+1)D nonlinear medium.

PubMed

Chen, Shihua; Soto-Crespo, Jose M; Baronio, Fabio; Grelu, Philippe; Mihalache, Dumitru

2016-07-11

We show that nonlinear wave packets localized in two dimensions with characteristic rogue wave profiles can propagate in a third dimension with significant stability. This unique behavior makes these waves analogous to light bullets, with the additional feature that they propagate on a finite background. Bulletlike rogue-wave singlet and triplet are derived analytically from a composite (2+1)D nonlinear wave equation. The latter can be interpreted as the combination of two integrable (1+1)D models expressed in different dimensions, namely, the Hirota equation and the complex modified Korteweg-de Vries equation. Numerical simulations confirm that the generation of rogue-wave bullets can be observed in the presence of spontaneous modulation instability activated by quantum noise.

Dayside Magnetosphere-Ionosphere Coupling and Prompt Response of Low-Latitude/Equatorial Ionosphere

NASA Astrophysics Data System (ADS)

Tu, J.; Song, P.

2017-12-01

We use a newly developed numerical simulation model of the ionosphere/thermosphere to investigate magnetosphere-ionosphere coupling and response of the low-latitude/equatorial ionosphere. The simulation model adapts an inductive-dynamic approach (including self-consistent solutions of Faraday's law and retaining inertia terms in ion momentum equations), that is, based on magnetic field B and plasma velocity v (B-v paradigm), in contrast to the conventional modeling based on electric field E and current j (E-j paradigm). The most distinct feature of this model is that the magnetic field in the ionosphere is not constant but self-consistently varies, e.g., with currents, in time. The model solves self-consistently time-dependent continuity, momentum, and energy equations for multiple species of ions and neutrals including photochemistry, and Maxwell's equations. The governing equations solved in the model are a set of multifluid-collisional-Hall MHD equations which are one of unique features of our ionosphere/thermosphere model. With such an inductive-dynamic approach, all possible MHD wave modes, each of which may refract and reflect depending on the local conditions, are retained in the solutions so that the dynamic coupling between the magnetosphere and ionosphere and among different regions of the ionosphere can be self-consistently investigated. In this presentation, we show that the disturbances propagate in the Alfven speed from the magnetosphere along the magnetic field lines down to the ionosphere/thermosphere and that they experience a mode conversion to compressional mode MHD waves (particularly fast mode) in the ionosphere. Because the fast modes can propagate perpendicular to the field, they propagate from the dayside high-latitude to the nightside as compressional waves and to the dayside low-latitude/equatorial ionosphere as rarefaction waves. The apparent prompt response of the low-latitude/equatorial ionosphere, manifesting as the sudden increase of the upward flow around the equator and global antisunward convection, is the result of such coupling of the high-latitude and the low-latitude/equatorial ionosphere, and the requirement of the flow continuity, instead of mechanisms such as the penetration electric field.

New approach to analyzing soil-building systems

USGS Publications Warehouse

Safak, E.

1998-01-01

A new method of analyzing seismic response of soil-building systems is introduced. The method is based on the discrete-time formulation of wave propagation in layered media for vertically propagating plane shear waves. Buildings are modeled as an extension of the layered soil media by assuming that each story in the building is another layer. The seismic response is expressed in terms of wave travel times between the layers, and the wave reflection and transmission coefficients at layer interfaces. The calculation of the response is reduced to a pair of simple finite-difference equations for each layer, which are solved recursively starting from the bedrock. Compared with commonly used vibration formulation, the wave propagation formulation provides several advantages, including the ability to incorporate soil layers, simplicity of the calculations, improved accuracy in modeling the mass and damping, and better tools for system identification and damage detection.A new method of analyzing seismic response of soil-building systems is introduced. The method is based on the discrete-time formulation of wave propagation in layered media for vertically propagating plane shear waves. Buildings are modeled as an extension of the layered soil media by assuming that each story in the building is another layer. The seismic response is expressed in terms of wave travel times between the layers, and the wave reflection and transmission coefficients at layer interfaces. The calculation of the response is reduced to a pair of simple finite-difference equations for each layer, which are solved recursively starting from the bedrock. Compared with commonly used vibration formulation, the wave propagation formulation provides several advantages, including the ability to incorporate soil layers, simplicity of the calculations, improved accuracy in modeling the mass and damping, and better tools for system identification and damage detection.

Remote recoil: a new wave mean interaction effect

NASA Astrophysics Data System (ADS)

Bühler, Oliver; McIntyre, Michael E.

2003-10-01

We present a theoretical study of a fundamentally new wave mean or wave vortex interaction effect able to force persistent, cumulative change in mean flows in the absence of wave breaking or other kinds of wave dissipation. It is associated with the refraction of non-dissipating waves by inhomogeneous mean (vortical) flows. The effect is studied in detail in the simplest relevant model, the two-dimensional compressible flow equations with a generic polytropic equation of state. This includes the usual shallow-water equations as a special case. The refraction of a narrow, slowly varying wavetrain of small-amplitude gravity or sound waves obliquely incident on a single weak (low Froude or Mach number) vortex is studied in detail. It is shown that, concomitant with the changes in the waves' pseudomomentum due to the refraction, there is an equal and opposite recoil force that is felt, in effect, by the vortex core. This effective force is called a ‘remote recoil’ to stress that there is no need for the vortex core and wavetrain to overlap in physical space. There is an accompanying ‘far-field recoil’ that is still more remote, as in classical vortex-impulse problems. The remote-recoil effects are studied perturbatively using the wave amplitude and vortex weakness as small parameters. The nature of the remote recoil is demonstrated in various set-ups with wavetrains of finite or infinite length. The effective recoil force {bm R}_V on the vortex core is given by an expression resembling the classical Magnus force felt by moving cylinders with circulation. In the case of wavetrains of infinite length, an explicit formula for the scattering angle theta_* of waves passing a vortex at a distance is derived correct to second order in Froude or Mach number. To this order {bm R}_V {~} theta_*. The formula is cross-checked against numerical integrations of the ray-tracing equations. This work is part of an ongoing study of internal-gravity-wave dynamics in the atmosphere and may be important for the development of future gravity-wave parametrization schemes in numerical models of the global atmospheric circulation. At present, all such schemes neglect remote-recoil effects caused by horizontally inhomogeneous mean flows. Taking these effects into account should make the parametrization schemes significantly more accurate.

Temporal evolutions and stationary waves for dissipative Benjamin-Ono equation

NASA Astrophysics Data System (ADS)

Feng, Bao-Feng; Kawahara, Takuji

2000-05-01

Initial value problems as well as stationary solitary and periodic waves are investigated for dissipative Benjamin-Ono (DBO) equation. Multi-hump stationary waves and their structures are identified numerically and the stability regions of stationary periodic waves are also examined numerically. These results elucidate a close relation between irregular behaviours in the initial value problem and the multiplicity of stationary waves.

Wave equation datuming applied to S-wave reflection seismic data

NASA Astrophysics Data System (ADS)

Tinivella, U.; Giustiniani, M.; Nicolich, R.

2018-05-01

S-wave high-resolution reflection seismic data was processed using Wave Equation Datuming technique in order to improve signal/noise ratio, attenuating coherent noise, and seismic resolution and to solve static corrections problems. The application of this algorithm allowed obtaining a good image of the shallow subsurface geological features. Wave Equation Datuming moves shots and receivers from a surface to another datum (the datum plane), removing time shifts originated by elevation variation and/or velocity changes in the shallow subsoil. This algorithm has been developed and currently applied to P wave, but it reveals the capacity to highlight S-waves images when used to resolve thin layers in high-resolution prospecting. A good S-wave image facilitates correlation with well stratigraphies, optimizing cost/benefit ratio of any drilling. The application of Wave Equation Datuming requires a reliable velocity field, so refraction tomography was adopted. The new seismic image highlights the details of the subsoil reflectors and allows an easier integration with borehole information and geological surveys than the seismic section obtained by conventional CMP reflection processing. In conclusion, the analysis of S-wave let to characterize the shallow subsurface recognizing levels with limited thickness once we have clearly attenuated ground roll, wind and environmental noise.

Non-autonomous multi-rogue waves for spin-1 coupled nonlinear Gross-Pitaevskii equation and management by external potentials.

PubMed

Li, Li; Yu, Fajun

2017-09-06

We investigate non-autonomous multi-rogue wave solutions in a three-component(spin-1) coupled nonlinear Gross-Pitaevskii(GP) equation with varying dispersions, higher nonlinearities, gain/loss and external potentials. The similarity transformation allows us to relate certain class of multi-rogue wave solutions of the spin-1 coupled nonlinear GP equation to the solutions of integrable coupled nonlinear Schrödinger(CNLS) equation. We study the effect of time-dependent quadratic potential on the profile and dynamic of non-autonomous rogue waves. With certain requirement on the backgrounds, some non-autonomous multi-rogue wave solutions exhibit the different shapes with two peaks and dip in bright-dark rogue waves. Then, the managements with external potential and dynamic behaviors of these solutions are investigated analytically. The results could be of interest in such diverse fields as Bose-Einstein condensates, nonlinear fibers and super-fluids.

A new mathematical approach for shock-wave solution in a dusty plasma

DOE Office of Scientific and Technical Information (OSTI.GOV)

Das, G.C.; Dwivedi, C.B.; Talukdar, M.

1997-12-01

The problem of nonlinear Burger equation in a plasma contaminated with heavy dust grains has been revisited. As discussed earlier [C. B. Dwivedi and B. P. Pandey, Phys. Plasmas {bold 2}, 9 (1995)], the Burger equation originates due to dust charge fluctuation dynamics. A new alternate mathematical approach based on a simple traveling wave formalism has been applied to find out the solution of the derived Burger equation, and the method recovers the known shock-wave solution. This technique, although having its own limitation, predicts successfully the salient features of the weak shock-wave structure in a dusty plasma with dust chargemore » fluctuation dynamics. It is emphasized that this approach of the traveling wave formalism is being applied for the first time to solve the nonlinear wave equation in plasmas. {copyright} {ital 1997 American Institute of Physics.}« less

A phase space approach to wave propagation with dispersion.

PubMed

Ben-Benjamin, Jonathan S; Cohen, Leon; Loughlin, Patrick J

2015-08-01

A phase space approximation method for linear dispersive wave propagation with arbitrary initial conditions is developed. The results expand on a previous approximation in terms of the Wigner distribution of a single mode. In contrast to this previously considered single-mode case, the approximation presented here is for the full wave and is obtained by a different approach. This solution requires one to obtain (i) the initial modal functions from the given initial wave, and (ii) the initial cross-Wigner distribution between different modal functions. The full wave is the sum of modal functions. The approximation is obtained for general linear wave equations by transforming the equations to phase space, and then solving in the new domain. It is shown that each modal function of the wave satisfies a Schrödinger-type equation where the equivalent "Hamiltonian" operator is the dispersion relation corresponding to the mode and where the wavenumber is replaced by the wavenumber operator. Application to the beam equation is considered to illustrate the approach.

Modulational instability and dynamics of implicit higher-order rogue wave solutions for the Kundu equation

NASA Astrophysics Data System (ADS)

Wen, Xiao-Yong; Zhang, Guoqiang

2018-01-01

Under investigation in this paper is the Kundu equation, which may be used to describe the propagation process of ultrashort optical pulses in nonlinear optics. The modulational instability of the plane-wave for the possible reason of the formation of the rogue wave (RW) is studied for the system. Based on our proposed generalized perturbation (n,N - n)-fold Darboux transformation (DT), some new higher-order implicit RW solutions in terms of determinants are obtained by means of the generalized perturbation (1,N - 1)-fold DT, when choosing different special parameters, these results will reduce to the RW solutions of the Kaup-Newell (KN) equation, Chen-Lee-Liu (CLL) equation and Gerjikov-Ivanov (GI) equation, respectively. The relevant wave structures are shown graphically, which display abundant interesting wave structures. The dynamical behaviors and propagation stability of the first-order and second-order RW solutions are discussed by using numerical simulations, the higher-order nonlinear terms for the Kundu equation have an impact on the propagation instability of the RW. The method can also be extended to find the higher-order RW or rational solutions of other integrable nonlinear equations.

A boundary integral approach to the scattering of nonplanar acoustic waves by rigid bodies

NASA Technical Reports Server (NTRS)

Gallman, Judith M.; Myers, M. K.; Farassat, F.

1990-01-01

The acoustic scattering of an incident wave by a rigid body can be described by a singular Fredholm integral equation of the second kind. This equation is derived by solving the wave equation using generalized function theory, Green's function for the wave equation in unbounded space, and the acoustic boundary condition for a perfectly rigid body. This paper will discuss the derivation of the wave equation, its reformulation as a boundary integral equation, and the solution of the integral equation by the Galerkin method. The accuracy of the Galerkin method can be assessed by applying the technique outlined in the paper to reproduce the known pressure fields that are due to various point sources. From the analysis of these simpler cases, the accuracy of the Galerkin solution can be inferred for the scattered pressure field caused by the incidence of a dipole field on a rigid sphere. The solution by the Galerkin technique can then be applied to such problems as a dipole model of a propeller whose pressure field is incident on a rigid cylinder. This is the groundwork for modeling the scattering of rotating blade noise by airplane fuselages.

Traveling wave solutions and conservation laws for nonlinear evolution equation

NASA Astrophysics Data System (ADS)

Baleanu, Dumitru; Inc, Mustafa; Yusuf, Abdullahi; Aliyu, Aliyu Isa

2018-02-01

In this work, the Riccati-Bernoulli sub-ordinary differential equation and modified tanh-coth methods are used to reach soliton solutions of the nonlinear evolution equation. We acquire new types of traveling wave solutions for the governing equation. We show that the equation is nonlinear self-adjoint by obtaining suitable substitution. Therefore, we construct conservation laws for the equation using new conservation theorem. The obtained solutions in this work may be used to explain and understand the physical nature of the wave spreads in the most dispersive medium. The constraint condition for the existence of solitons is stated. Some three dimensional figures for some of the acquired results are illustrated.

Diffusion phenomenon for linear dissipative wave equations in an exterior domain

NASA Astrophysics Data System (ADS)

Ikehata, Ryo

Under the general condition of the initial data, we will derive the crucial estimates which imply the diffusion phenomenon for the dissipative linear wave equations in an exterior domain. In order to derive the diffusion phenomenon for dissipative wave equations, the time integral method which was developed by Ikehata and Matsuyama (Sci. Math. Japon. 55 (2002) 33) plays an effective role.

Nonlinear Waves and Inverse Scattering

DTIC Science & Technology

1990-09-18

to be published Proceedings: conference Chaos in Australia (February 1990). 5. On the Kadomtsev Petviashvili Equation and Associated Constraints by...Scattering Transfoni (IST). IST is a method which alows one to’solve nonlinear wave equations by solving certain related direct and inverse scattering...problems. We use these results to find solutions to nonlinear wave equations much like one uses Fourier analysis for linear problems. Moreover the

Inverse scattering pre-stack depth imaging and it's comparison to some depth migration methods for imaging rich fault complex structure

NASA Astrophysics Data System (ADS)

Nurhandoko, Bagus Endar B.; Sukmana, Indriani; Mubarok, Syahrul; Deny, Agus; Widowati, Sri; Kurniadi, Rizal

2012-06-01

Migration is important issue for seismic imaging in complex structure. In this decade, depth imaging becomes important tools for producing accurate image in depth imaging instead of time domain imaging. The challenge of depth migration method, however, is in revealing the complex structure of subsurface. There are many methods of depth migration with their advantages and weaknesses. In this paper, we show our propose method of pre-stack depth migration based on time domain inverse scattering wave equation. Hopefully this method can be as solution for imaging complex structure in Indonesia, especially in rich thrusting fault zones. In this research, we develop a recent advance wave equation migration based on time domain inverse scattering wave which use more natural wave propagation using scattering wave. This wave equation pre-stack depth migration use time domain inverse scattering wave equation based on Helmholtz equation. To provide true amplitude recovery, an inverse of divergence procedure and recovering transmission loss are considered of pre-stack migration. Benchmarking the propose inverse scattering pre-stack depth migration with the other migration methods are also presented, i.e.: wave equation pre-stack depth migration, waveequation depth migration, and pre-stack time migration method. This inverse scattering pre-stack depth migration could image successfully the rich fault zone which consist extremely dip and resulting superior quality of seismic image. The image quality of inverse scattering migration is much better than the others migration methods.

The quantum universe

NASA Astrophysics Data System (ADS)

Hey, Anthony J. G.; Walters, Patrick

This book provides a descriptive, popular account of quantum physics. The basic topics addressed include: waves and particles, the Heisenberg uncertainty principle, the Schroedinger equation and matter waves, atoms and nuclei, quantum tunneling, the Pauli exclusion principle and the elements, quantum cooperation and superfluids, Feynman rules, weak photons, quarks, and gluons. The applications of quantum physics to astrophyics, nuclear technology, and modern electronics are addressed.

Influence of optical activity on rogue waves propagating in chiral optical fibers.

PubMed

Temgoua, D D Estelle; Kofane, T C

2016-06-01

We derive the nonlinear Schrödinger (NLS) equation in chiral optical fiber with right- and left-hand nonlinear polarization. We use the similarity transformation to reduce the generalized chiral NLS equation to the higher-order integrable Hirota equation. We present the first- and second-order rational solutions of the chiral NLS equation with variable and constant coefficients, based on the modified Darboux transformation method. For some specific set of parameters, the features of chiral optical rogue waves are analyzed from analytical results, showing the influence of optical activity on waves. We also generate the exact solutions of the two-component coupled nonlinear Schrödinger equations, which describe optical activity effects on the propagation of rogue waves, and their properties in linear and nonlinear coupling cases are investigated. The condition of modulation instability of the background reveals the existence of vector rogue waves and the number of stable and unstable branches. Controllability of chiral optical rogue waves is examined by numerical simulations and may bring potential applications in optical fibers and in many other physical systems.

Data-driven discovery of partial differential equations

PubMed Central

Rudy, Samuel H.; Brunton, Steven L.; Proctor, Joshua L.; Kutz, J. Nathan

2017-01-01

We propose a sparse regression method capable of discovering the governing partial differential equation(s) of a given system by time series measurements in the spatial domain. The regression framework relies on sparsity-promoting techniques to select the nonlinear and partial derivative terms of the governing equations that most accurately represent the data, bypassing a combinatorially large search through all possible candidate models. The method balances model complexity and regression accuracy by selecting a parsimonious model via Pareto analysis. Time series measurements can be made in an Eulerian framework, where the sensors are fixed spatially, or in a Lagrangian framework, where the sensors move with the dynamics. The method is computationally efficient, robust, and demonstrated to work on a variety of canonical problems spanning a number of scientific domains including Navier-Stokes, the quantum harmonic oscillator, and the diffusion equation. Moreover, the method is capable of disambiguating between potentially nonunique dynamical terms by using multiple time series taken with different initial data. Thus, for a traveling wave, the method can distinguish between a linear wave equation and the Korteweg–de Vries equation, for instance. The method provides a promising new technique for discovering governing equations and physical laws in parameterized spatiotemporal systems, where first-principles derivations are intractable. PMID:28508044

Ginzburg-Landau equation as a heuristic model for generating rogue waves

NASA Astrophysics Data System (ADS)

Lechuga, Antonio

2016-04-01

Envelope equations have many applications in the study of physical systems. Particularly interesting is the case 0f surface water waves. In steady conditions, laboratory experiments are carried out for multiple purposes either for researches or for practical problems. In both cases envelope equations are useful for understanding qualitative and quantitative results. The Ginzburg-Landau equation provides an excellent model for systems of that kind with remarkable patterns. Taking into account the above paragraph the main aim of our work is to generate waves in a water tank with almost a symmetric spectrum according to Akhmediev (2011) and thus, to produce a succession of rogue waves. The envelope of these waves gives us some patterns whose model is a type of Ginzburg-Landau equation, Danilov et al (1988). From a heuristic point of view the link between the experiment and the model is achieved. Further, the next step consists of changing generating parameters on the water tank and also the coefficients of the Ginzburg-Landau equation, Lechuga (2013) in order to reach a sufficient good approach.

Application of a Phase-resolving, Directional Nonlinear Spectral Wave Model

NASA Astrophysics Data System (ADS)

Davis, J. R.; Sheremet, A.; Tian, M.; Hanson, J. L.

2014-12-01

We describe several applications of a phase-resolving, directional nonlinear spectral wave model. The model describes a 2D surface gravity wave field approaching a mildly sloping beach with parallel depth contours at an arbitrary angle accounting for nonlinear, quadratic triad interactions. The model is hyperbolic, with the initial wave spectrum specified in deep water. Complex amplitudes are generated based on the random phase approximation. The numerical implementation includes unidirectional propagation as a special case. In directional mode, it solves the system of equations in the frequency-alongshore wave number space. Recent enhancements of the model include the incorporation of dissipation caused by breaking and propagation over a viscous mud layer and the calculation of wave induced setup. Applications presented include: a JONSWAP spectrum with a cos2s directional distribution, for shore-perpendicular and oblique propagation, a study of the evolution of a single directional triad, and several preliminary comparisons to wave spectra collected at the USACE-FRF in Duck, NC which show encouraging results although further validation with a wider range of beach slopes and wave conditions is needed.

Explicit Solutions and Bifurcations for a Class of Generalized Boussinesq Wave Equation

NASA Astrophysics Data System (ADS)

Ma, Zhi-Min; Sun, Yu-Huai; Liu, Fu-Sheng

2013-03-01

In this paper, the generalized Boussinesq wave equation utt — uxx + a(um)xx + buxxxx = 0 is investigated by using the bifurcation theory and the method of phase portraits analysis. Under the different parameter conditions, the exact explicit parametric representations for solitary wave solutions and periodic wave solutions are obtained.

Modelling rogue waves through exact dynamical lump soliton controlled by ocean currents.

PubMed

Kundu, Anjan; Mukherjee, Abhik; Naskar, Tapan

2014-04-08

Rogue waves are extraordinarily high and steep isolated waves, which appear suddenly in a calm sea and disappear equally fast. However, though the rogue waves are localized surface waves, their theoretical models and experimental observations are available mostly in one dimension, with the majority of them admitting only limited and fixed amplitude and modular inclination of the wave. We propose two dimensions, exactly solvable nonlinear Schrödinger (NLS) equation derivable from the basic hydrodynamic equations and endowed with integrable structures. The proposed two-dimensional equation exhibits modulation instability and frequency correction induced by the nonlinear effect, with a directional preference, all of which can be determined through precise analytic result. The two-dimensional NLS equation allows also an exact lump soliton which can model a full-grown surface rogue wave with adjustable height and modular inclination. The lump soliton under the influence of an ocean current appears and disappears preceded by a hole state, with its dynamics controlled by the current term. These desirable properties make our exact model promising for describing ocean rogue waves.

Modelling rogue waves through exact dynamical lump soliton controlled by ocean currents

PubMed Central

Kundu, Anjan; Mukherjee, Abhik; Naskar, Tapan

2014-01-01

Rogue waves are extraordinarily high and steep isolated waves, which appear suddenly in a calm sea and disappear equally fast. However, though the rogue waves are localized surface waves, their theoretical models and experimental observations are available mostly in one dimension, with the majority of them admitting only limited and fixed amplitude and modular inclination of the wave. We propose two dimensions, exactly solvable nonlinear Schrödinger (NLS) equation derivable from the basic hydrodynamic equations and endowed with integrable structures. The proposed two-dimensional equation exhibits modulation instability and frequency correction induced by the nonlinear effect, with a directional preference, all of which can be determined through precise analytic result. The two-dimensional NLS equation allows also an exact lump soliton which can model a full-grown surface rogue wave with adjustable height and modular inclination. The lump soliton under the influence of an ocean current appears and disappears preceded by a hole state, with its dynamics controlled by the current term. These desirable properties make our exact model promising for describing ocean rogue waves. PMID:24711719

Testing the Perey effect

DOE PAGES

Titus, L. J.; Nunes, Filomena M.

2014-03-12

Here, the effects of non-local potentials have historically been approximately included by applying a correction factor to the solution of the corresponding equation for the local equivalent interaction. This is usually referred to as the Perey correction factor. In this work we investigate the validity of the Perey correction factor for single-channel bound and scattering states, as well as in transfer (p, d) cross sections. Method: We solve the scattering and bound state equations for non-local interactions of the Perey-Buck type, through an iterative method. Using the distorted wave Born approximation, we construct the T-matrix for (p,d) on 17O, 41Ca,more » 49Ca, 127Sn, 133Sn, and 209Pb at 20 and 50 MeV. As a result, we found that for bound states, the Perey corrected wave function resulting from the local equation agreed well with that from the non-local equation in the interior region, but discrepancies were found in the surface and peripheral regions. Overall, the Perey correction factor was adequate for scattering states, with the exception of a few partial waves corresponding to the grazing impact parameters. These differences proved to be important for transfer reactions. In conclusion, the Perey correction factor does offer an improvement over taking a direct local equivalent solution. However, if the desired accuracy is to be better than 10%, the exact solution of the non-local equation should be pursued.« less

Solitary-wave solutions of the Benjamin equation

DOE Office of Scientific and Technical Information (OSTI.GOV)

Albert, J.P.; Bona, J.L.; Restrepo, J.M.

1999-10-01

Considered here is a model equation put forward by Benjamin that governs approximately the evolution of waves on the interface of a two-fluid system in which surface-tension effects cannot be ignored. The principal focus is the traveling-wave solutions called solitary waves, and three aspects will be investigated. A constructive proof of the existence of these waves together with a proof of their stability is developed. Continuation methods are used to generate a scheme capable of numerically approximating these solitary waves. The computer-generated approximations reveal detailed aspects of the structure of these waves. They are symmetric about their crests, but unlikemore » the classical Korteqeg-de Vries solitary waves, they feature a finite number of oscillations. The derivation of the equation is also revisited to get an idea of whether or not these oscillatory waves might actually occur in a natural setting.« less

A single-sided representation for the homogeneous Green's function of a unified scalar wave equation.

PubMed

Wapenaar, Kees

2017-06-01

A unified scalar wave equation is formulated, which covers three-dimensional (3D) acoustic waves, 2D horizontally-polarised shear waves, 2D transverse-electric EM waves, 2D transverse-magnetic EM waves, 3D quantum-mechanical waves and 2D flexural waves. The homogeneous Green's function of this wave equation is a combination of the causal Green's function and its time-reversal, such that their singularities at the source position cancel each other. A classical representation expresses this homogeneous Green's function as a closed boundary integral. This representation finds applications in holographic imaging, time-reversed wave propagation and Green's function retrieval by cross correlation. The main drawback of the classical representation in those applications is that it requires access to a closed boundary around the medium of interest, whereas in many practical situations the medium can be accessed from one side only. Therefore, a single-sided representation is derived for the homogeneous Green's function of the unified scalar wave equation. Like the classical representation, this single-sided representation fully accounts for multiple scattering. The single-sided representation has the same applications as the classical representation, but unlike the classical representation it is applicable in situations where the medium of interest is accessible from one side only.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Menikoff, Ralph

The Zel’dovich-von Neumann-Doering (ZND) profile of a detonation wave is derived. Two basic assumptions are required: i. An equation of state (EOS) for a partly burned explosive; P(V, e, λ). ii. A burn rate for the reaction progress variable; d/dt λ = R(V, e, λ). For a steady planar detonation wave the reactive flow PDEs can be reduced to ODEs. The detonation wave profile can be determined from an ODE plus algebraic equations for points on the partly burned detonation loci with a specified wave speed. Furthermore, for the CJ detonation speed the end of the reaction zone is sonic.more » A solution to the reactive flow equations can be constructed with a rarefaction wave following the detonation wave profile. This corresponds to an underdriven detonation wave, and the rarefaction is know as a Taylor wave.« less

On the solution of the generalized wave and generalized sine-Gordon equations

NASA Technical Reports Server (NTRS)

Ablowitz, M. J.; Beals, R.; Tenenblat, K.

1986-01-01

The generalized wave equation and generalized sine-Gordon equations are known to be natural multidimensional differential geometric generalizations of the classical two-dimensional versions. In this paper, a system of linear differential equations is associated with these equations, and it is shown how the direct and inverse problems can be solved for appropriately decaying data on suitable lines. An initial-boundary value problem is solved for these equations.

Two-dimensional evolution equation of finite-amplitude internal gravity waves in a uniformly stratified fluid

PubMed

Kataoka; Tsutahara; Akuzawa

2000-02-14

We derive a fully nonlinear evolution equation that can describe the two-dimensional motion of finite-amplitude long internal waves in a uniformly stratified three-dimensional fluid of finite depth. The derived equation is the two-dimensional counterpart of the evolution equation obtained by Grimshaw and Yi [J. Fluid Mech. 229, 603 (1991)]. In the small-amplitude limit, our equation is reduced to the celebrated Kadomtsev-Petviashvili equation.

Twisted rogue-wave pairs in the Sasa-Satsuma equation.

PubMed

Chen, Shihua

2013-08-01

Exact explicit rogue wave solutions of the Sasa-Satsuma equation are obtained by use of a Darboux transformation. In addition to the double-peak structure and an analog of the Peregrine soliton, the rogue wave can exhibit an intriguing twisted rogue-wave pair that involves four well-defined zero-amplitude points. This exotic structure may enrich our understanding on the nature of rogue waves.

Influence of a weak gravitational wave on a bound system of two point-masses. [of binary stars

NASA Technical Reports Server (NTRS)

Turner, M. S.

1979-01-01

The problem of a weak gravitational wave impinging upon a nonrelativistic bound system of two point masses is considered. The geodesic equation for each mass is expanded in terms of two small parameters, v/c and dimensionless wave amplitude, in a manner similar to the post-Newtonian expansion; the geodesic equations are resolved into orbital and center-of-mass equations of motion. The effect of the wave on the orbit is determined by using Lagrange's planetary equations to calculate the time evolution of the orbital elements. The gauge properties of the solutions and, in particular, the gauge invariance of the secular effects are discussed.

Transport equations for linear surface waves with random underlying flows

NASA Astrophysics Data System (ADS)

Bal, Guillaume; Chou, Tom

1999-11-01

We define the Wigner distribution and use it to develop equations for linear surface capillary-gravity wave propagation in the transport regime. The energy density a(r, k) contained in waves propagating with wavevector k at field point r is given by dota(r,k) + nabla_k[U_⊥(r,z=0) \\cdotk + Ω(k)]\\cdotnabla_ra [13pt] \\: hspace1in - (nabla_r\\cdotU_⊥)a - nabla_r(k\\cdotU_⊥)\\cdotnabla_ka = Σ(δU^2) where U_⊥(r, z=0) is a slowly varying surface current, and Ω(k) = √(k^3+k)tanh kh is the free capillary-gravity dispersion relation. Note that nabla_r\\cdotU_⊥(r,z=0) neq 0, and that the surface currents exchange energy density with the propagating waves. When an additional weak random current √\\varepsilon δU(r/\\varepsilon) varying on the scale of k-1 is included, we find an additional scattering term Σ(δU^2) as a function of correlations in δU. Our results can be applied to the study of surface wave energy transport over a turbulent ocean.

A Wideband Fast Multipole Method for the two-dimensional complex Helmholtz equation

NASA Astrophysics Data System (ADS)

Cho, Min Hyung; Cai, Wei

2010-12-01

A Wideband Fast Multipole Method (FMM) for the 2D Helmholtz equation is presented. It can evaluate the interactions between N particles governed by the fundamental solution of 2D complex Helmholtz equation in a fast manner for a wide range of complex wave number k, which was not easy with the original FMM due to the instability of the diagonalized conversion operator. This paper includes the description of theoretical backgrounds, the FMM algorithm, software structures, and some test runs. Program summaryProgram title: 2D-WFMM Catalogue identifier: AEHI_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEHI_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 4636 No. of bytes in distributed program, including test data, etc.: 82 582 Distribution format: tar.gz Programming language: C Computer: Any Operating system: Any operating system with gcc version 4.2 or newer Has the code been vectorized or parallelized?: Multi-core processors with shared memory RAM: Depending on the number of particles N and the wave number k Classification: 4.8, 4.12 External routines: OpenMP ( http://openmp.org/wp/) Nature of problem: Evaluate interaction between N particles governed by the fundamental solution of 2D Helmholtz equation with complex k. Solution method: Multilevel Fast Multipole Algorithm in a hierarchical quad-tree structure with cutoff level which combines low frequency method and high frequency method. Running time: Depending on the number of particles N, wave number k, and number of cores in CPU. CPU time increases as N log N.

On the evolution of perturbations to solutions of the Kadomtsev-Petviashvilli equation using the Benney-Luke equation

NASA Astrophysics Data System (ADS)

Ablowitz, Mark J.; Curtis, Christopher W.

2011-05-01

The Benney-Luke equation, which arises as a long wave asymptotic approximation of water waves, contains the Kadomtsev-Petviashvilli (KP) equation as a leading-order maximal balanced approximation. The question analyzed is how the Benney-Luke equation modifies the so-called web solutions of the KP equation. It is found that the Benney-Luke equation introduces dispersive radiation which breaks each of the symmetric soliton-like humps well away from the interaction region of the KP web solution into a tail of multi-peaked oscillating profiles behind the main solitary hump. Computation indicates that the wave structure is modified near the center of the interaction region. Both analytical and numerical techniques are employed for working with non-periodic, non-decaying solutions on unbounded domains.

Exact solutions of unsteady Korteweg-de Vries and time regularized long wave equations.

PubMed

Islam, S M Rayhanul; Khan, Kamruzzaman; Akbar, M Ali

2015-01-01

In this paper, we implement the exp(-Φ(ξ))-expansion method to construct the exact traveling wave solutions for nonlinear evolution equations (NLEEs). Here we consider two model equations, namely the Korteweg-de Vries (KdV) equation and the time regularized long wave (TRLW) equation. These equations play significant role in nonlinear sciences. We obtained four types of explicit function solutions, namely hyperbolic, trigonometric, exponential and rational function solutions of the variables in the considered equations. It has shown that the applied method is quite efficient and is practically well suited for the aforementioned problems and so for the other NLEEs those arise in mathematical physics and engineering fields. PACS numbers: 02.30.Jr, 02.70.Wz, 05.45.Yv, 94.05.Fq.

Computational Modeling of Ultrafast Pulse Propagation in Nonlinear Optical Materials

NASA Technical Reports Server (NTRS)

Goorjian, Peter M.; Agrawal, Govind P.; Kwak, Dochan (Technical Monitor)

1996-01-01

There is an emerging technology of photonic (or optoelectronic) integrated circuits (PICs or OEICs). In PICs, optical and electronic components are grown together on the same chip. rib build such devices and subsystems, one needs to model the entire chip. Accurate computer modeling of electromagnetic wave propagation in semiconductors is necessary for the successful development of PICs. More specifically, these computer codes would enable the modeling of such devices, including their subsystems, such as semiconductor lasers and semiconductor amplifiers in which there is femtosecond pulse propagation. Here, the computer simulations are made by solving the full vector, nonlinear, Maxwell's equations, coupled with the semiconductor Bloch equations, without any approximations. The carrier is retained in the description of the optical pulse, (i.e. the envelope approximation is not made in the Maxwell's equations), and the rotating wave approximation is not made in the Bloch equations. These coupled equations are solved to simulate the propagation of femtosecond optical pulses in semiconductor materials. The simulations describe the dynamics of the optical pulses, as well as the interband and intraband.

On critical behaviour in generalized Kadomtsev-Petviashvili equations

NASA Astrophysics Data System (ADS)

Dubrovin, B.; Grava, T.; Klein, C.

2016-10-01

An asymptotic description of the formation of dispersive shock waves in solutions to the generalized Kadomtsev-Petviashvili (KP) equation is conjectured. The asymptotic description based on a multiscales expansion is given in terms of a special solution to an ordinary differential equation of the Painlevé I hierarchy. Several examples are discussed numerically to provide strong evidence for the validity of the conjecture. The numerical study of the long time behaviour of these examples indicates persistence of dispersive shock waves in solutions to the (subcritical) KP equations, while in the supercritical KP equations a blow-up occurs after the formation of the dispersive shock waves.

The Weyl-Lanczos equations and the Lanczos wave equation in four dimensions as systems in involution

NASA Astrophysics Data System (ADS)

Dolan, P.; Gerber, A.

2003-07-01

The Weyl-Lanczos equations in four dimensions form a system in involution. We compute its Cartan characters explicitly and use Janet-Riquier theory to confirm the results in the case of all space-times with a diagonal metric tensor and for the plane wave limit of space-times. We write the Lanczos wave equation as an exterior differential system and, with assistance from Janet-Riquier theory, we compute its Cartan characters and find that it forms a system in involution. We compare these Cartan characters with those of the Weyl-Lanczos equations. All results hold for the real analytic case.

The propagation of the shock wave from a strong explosion in a plane-parallel stratified medium: the Kompaneets approximation

NASA Astrophysics Data System (ADS)

Olano, C. A.

2009-11-01

Context: Using certain simplifications, Kompaneets derived a partial differential equation that states the local geometrical and kinematical conditions that each surface element of a shock wave, created by a point blast in a stratified gaseous medium, must satisfy. Kompaneets could solve his equation analytically for the case of a wave propagating in an exponentially stratified medium, obtaining the form of the shock front at progressive evolutionary stages. Complete analytical solutions of the Kompaneets equation for shock wave motion in further plane-parallel stratified media were not found, except for radially stratified media. Aims: We aim to analytically solve the Kompaneets equation for the motion of a shock wave in different plane-parallel stratified media that can reflect a wide variety of astrophysical contexts. We were particularly interested in solving the Kompaneets equation for a strong explosion in the interstellar medium of the Galactic disk, in which, due to intense winds and explosions of stars, gigantic gaseous structures known as superbubbles and supershells are formed. Methods: Using the Kompaneets approximation, we derived a pair of equations that we call adapted Kompaneets equations, that govern the propagation of a shock wave in a stratified medium and that permit us to obtain solutions in parametric form. The solutions provided by the system of adapted Kompaneets equations are equivalent to those of the Kompaneets equation. We solved the adapted Kompaneets equations for shock wave propagation in a generic stratified medium by means of a power-series method. Results: Using the series solution for a shock wave in a generic medium, we obtained the series solutions for four specific media whose respective density distributions in the direction perpendicular to the stratification plane are of an exponential, power-law type (one with exponent k=-1 and the other with k =-2) and a quadratic hyperbolic-secant. From these series solutions, we deduced exact solutions for the four media in terms of elemental functions. The exact solution for shock wave propagation in a medium of quadratic hyperbolic-secant density distribution is very appropriate to describe the growth of superbubbles in the Galactic disk. Member of the Carrera del Investigador Científico del CONICET, Argentina.

New exact periodic solitary-wave solutions for the new (3+1)-dimensional generalized Kadomtsev-Petviashvili equation in multi-temperature electron plasmas

NASA Astrophysics Data System (ADS)

Liu, Jian-Guo; Tian, Yu; Zeng, Zhi-Fang

2017-10-01

In this paper, we aim to introduce a new form of the (3+1)-dimensional generalized Kadomtsev-Petviashvili equation for the long waves of small amplitude with slow dependence on the transverse coordinate. By using the Hirota's bilinear form and the extended homoclinic test approach, new exact periodic solitary-wave solutions for the new (3+1)-dimensional generalized Kadomtsev-Petviashvili equation are presented. Moreover, the properties and characteristics for these new exact periodic solitary-wave solutions are discussed with some figures.

Travelling wave solutions and conservation laws for the Korteweg-de Vries-Bejamin-Bona-Mahony equation

NASA Astrophysics Data System (ADS)

Simbanefayi, Innocent; Khalique, Chaudry Masood

2018-03-01

In this work we study the Korteweg-de Vries-Benjamin-Bona-Mahony (KdV-BBM) equation, which describes the two-way propagation of waves. Using Lie symmetry method together with Jacobi elliptic function expansion and Kudryashov methods we construct its travelling wave solutions. Also, we derive conservation laws of the KdV-BBM equation using the variational derivative approach. In this method, we begin by computing second-order multipliers for the KdV-BBM equation followed by a derivation of the respective conservation laws for each multiplier.

Analytical studies on the Benney-Luke equation in mathematical physics

NASA Astrophysics Data System (ADS)

Islam, S. M. Rayhanul; Khan, Kamruzzaman; Woadud, K. M. Abdul Al

2018-04-01

The enhanced (G‧/G)-expansion method presents wide applicability to handling nonlinear wave equations. In this article, we find the new exact traveling wave solutions of the Benney-Luke equation by using the enhanced (G‧/G)-expansion method. This method is a useful, reliable, and concise method to easily solve the nonlinear evaluation equations (NLEEs). The traveling wave solutions have expressed in term of the hyperbolic and trigonometric functions. We also have plotted the 2D and 3D graphics of some analytical solutions obtained in this paper.

A nonlinear wave equation in nonadiabatic flame propagation

DOE Office of Scientific and Technical Information (OSTI.GOV)

Booty, M.R.; Matalon, M.; Matkowsky, B.J.

1988-06-01

The authors derive a nonlinear wave equation from the diffusional thermal model of gaseous combustion to describe the evolution of a flame front. The equation arises as a long wave theory, for values of the volumeric heat loss in a neighborhood of the extinction point (beyond which planar uniformly propagating flames cease to exist), and for Lewis numbers near the critical value beyond which uniformly propagating planar flames lose stability via a degenerate Hopf bifurcation. Analysis of the equation suggests the possibility of a singularity developing in finite time.

On Richardson extrapolation for low-dissipation low-dispersion diagonally implicit Runge-Kutta schemes

NASA Astrophysics Data System (ADS)

Havasi, Ágnes; Kazemi, Ehsan

2018-04-01

In the modeling of wave propagation phenomena it is necessary to use time integration methods which are not only sufficiently accurate, but also properly describe the amplitude and phase of the propagating waves. It is not clear if amending the developed schemes by extrapolation methods to obtain a high order of accuracy preserves the qualitative properties of these schemes in the perspective of dissipation, dispersion and stability analysis. It is illustrated that the combination of various optimized schemes with Richardson extrapolation is not optimal for minimal dissipation and dispersion errors. Optimized third-order and fourth-order methods are obtained, and it is shown that the proposed methods combined with Richardson extrapolation result in fourth and fifth orders of accuracy correspondingly, while preserving optimality and stability. The numerical applications include the linear wave equation, a stiff system of reaction-diffusion equations and the nonlinear Euler equations with oscillatory initial conditions. It is demonstrated that the extrapolated third-order scheme outperforms the recently developed fourth-order diagonally implicit Runge-Kutta scheme in terms of accuracy and stability.

Applications of He's semi-inverse method, ITEM and GGM to the Davey-Stewartson equation

NASA Astrophysics Data System (ADS)

Zinati, Reza Farshbaf; Manafian, Jalil

2017-04-01

We investigate the Davey-Stewartson (DS) equation. Travelling wave solutions were found. In this paper, we demonstrate the effectiveness of the analytical methods, namely, He's semi-inverse variational principle method (SIVPM), the improved tan(φ/2)-expansion method (ITEM) and generalized G'/G-expansion method (GGM) for seeking more exact solutions via the DS equation. These methods are direct, concise and simple to implement compared to other existing methods. The exact solutions containing four types solutions have been achieved. The results demonstrate that the aforementioned methods are more efficient than the Ansatz method applied by Mirzazadeh (2015). Abundant exact travelling wave solutions including solitons, kink, periodic and rational solutions have been found by the improved tan(φ/2)-expansion and generalized G'/G-expansion methods. By He's semi-inverse variational principle we have obtained dark and bright soliton wave solutions. Also, the obtained semi-inverse variational principle has profound implications in physical understandings. These solutions might play important role in engineering and physics fields. Moreover, by using Matlab, some graphical simulations were done to see the behavior of these solutions.

Evolution of a proto-neutron star with a nuclear many-body equation of state: Neutrino luminosity and gravitational wave frequencies

DOE PAGES

Camelio, Giovanni; Lovato, Alessandro; Gualtieri, Leonardo; ...

2017-08-30

In a core-collapse supernova, a huge amount of energy is released in the Kelvin-Helmholtz phase subsequent to the explosion, when the proto-neutron star cools and deleptonizes as it loses neutrinos. Most of this energy is emitted through neutrinos, but a fraction of it can be released through gravitational waves. We model the evolution of a proto-neutron star in the Kelvin-Helmholtz phase using a general relativistic numerical code, and a recently proposed finite temperature, many-body equation of state; from this we consistently compute the diffusion coefficients driving the evolution. To include the many-body equation of state, we develop a new fittingmore » formula for the high density baryon free energy at finite temperature and intermediate proton fraction. Here, we estimate the emitted neutrino signal, assessing its detectability by present terrestrial detectors, and we determine the frequencies and damping times of the quasinormal modes which would characterize the gravitational wave signal emitted in this stage.« less

Evolution of a proto-neutron star with a nuclear many-body equation of state: Neutrino luminosity and gravitational wave frequencies

DOE Office of Scientific and Technical Information (OSTI.GOV)

Camelio, Giovanni; Lovato, Alessandro; Gualtieri, Leonardo

In a core-collapse supernova, a huge amount of energy is released in the Kelvin-Helmholtz phase subsequent to the explosion, when the proto-neutron star cools and deleptonizes as it loses neutrinos. Most of this energy is emitted through neutrinos, but a fraction of it can be released through gravitational waves. We model the evolution of a proto-neutron star in the Kelvin-Helmholtz phase using a general relativistic numerical code, and a recently proposed finite temperature, many-body equation of state; from this we consistently compute the diffusion coefficients driving the evolution. To include the many-body equation of state, we develop a new fittingmore » formula for the high density baryon free energy at finite temperature and intermediate proton fraction. Here, we estimate the emitted neutrino signal, assessing its detectability by present terrestrial detectors, and we determine the frequencies and damping times of the quasinormal modes which would characterize the gravitational wave signal emitted in this stage.« less

Full Wave Parallel Code for Modeling RF Fields in Hot Plasmas

NASA Astrophysics Data System (ADS)

Spencer, Joseph; Svidzinski, Vladimir; Evstatiev, Evstati; Galkin, Sergei; Kim, Jin-Soo

2015-11-01

FAR-TECH, Inc. is developing a suite of full wave RF codes in hot plasmas. It is based on a formulation in configuration space with grid adaptation capability. The conductivity kernel (which includes a nonlocal dielectric response) is calculated by integrating the linearized Vlasov equation along unperturbed test particle orbits. For Tokamak applications a 2-D version of the code is being developed. Progress of this work will be reported. This suite of codes has the following advantages over existing spectral codes: 1) It utilizes the localized nature of plasma dielectric response to the RF field and calculates this response numerically without approximations. 2) It uses an adaptive grid to better resolve resonances in plasma and antenna structures. 3) It uses an efficient sparse matrix solver to solve the formulated linear equations. The linear wave equation is formulated using two approaches: for cold plasmas the local cold plasma dielectric tensor is used (resolving resonances by particle collisions), while for hot plasmas the conductivity kernel is calculated. Work is supported by the U.S. DOE SBIR program.

Linear Elastic Waves - Series: Cambridge Texts in Applied Mathematics (No. 26)

NASA Astrophysics Data System (ADS)

Harris, John G.

2001-10-01

Wave propagation and scattering are among the most fundamental processes that we use to comprehend the world around us. While these processes are often very complex, one way to begin to understand them is to study wave propagation in the linear approximation. This is a book describing such propagation using, as a context, the equations of elasticity. Two unifying themes are used. The first is that an understanding of plane wave interactions is fundamental to understanding more complex wave interactions. The second is that waves are best understood in an asymptotic approximation where they are free of the complications of their excitation and are governed primarily by their propagation environments. The topics covered include reflection, refraction, the propagation of interfacial waves, integral representations, radiation and diffraction, and propagation in closed and open waveguides. Linear Elastic Waves is an advanced level textbook directed at applied mathematicians, seismologists, and engineers. Aimed at beginning graduate students Includes examples and exercises Has application in a wide range of disciplines

Solitons and SeaSat,

DTIC Science & Technology

1984-08-01

the Kadomtsev - • . Petviashvili (1) equations . A derivation of Eq. (1) in the case of . " * internal waves is given in reference (2). An important...second statement is demonstrated to be false. The% Kadomtsev -.1etviashvile equation relevant to Internal Waves is shown not to have SOliL -solutions. This...more than one space dimension. The second statement is demonstrated to be false. The Kadomtsev -Petviashvile equation relevant to Internal Waves Is

Wave propagation through a flexoelectric piezoelectric slab sandwiched by two piezoelectric half-spaces.

PubMed

Jiao, Fengyu; Wei, Peijun; Li, Yueqiu

2018-01-01

Reflection and transmission of plane waves through a flexoelectric piezoelectric slab sandwiched by two piezoelectric half-spaces are studied in this paper. The secular equations in the flexoelectric piezoelectric material are first derived from the general governing equation. Different from the classical piezoelectric medium, there are five kinds of coupled elastic waves in the piezoelectric material with the microstructure effects taken into consideration. The state vectors are obtained by the summation of contributions from all possible partial waves. The state transfer equation of flexoelectric piezoelectric slab is derived from the motion equation by the reduction of order, and the transfer matrix of flexoelectric piezoelectric slab is obtained by solving the state transfer equation. By using the continuous conditions at the interface and the approach of partition matrix, we get the resultant algebraic equations in term of the transfer matrix from which the reflection and transmission coefficients can be calculated. The amplitude ratios and further the energy flux ratios of various waves are evaluated numerically. The numerical results are shown graphically and are validated by the energy conservation law. Based on these numerical results, the influences of two characteristic lengths of microstructure and the flexoelectric coefficients on the wave propagation are discussed. Copyright © 2017 Elsevier B.V. All rights reserved.

Microscopic Lagrangian description of warm plasmas. I - Linear wave propagation. II - Nonlinear wave interactions

NASA Technical Reports Server (NTRS)

Kim, H.; Crawford, F. W.

1977-01-01

It is pointed out that the conventional iterative analysis of nonlinear plasma wave phenomena, which involves a direct use of Maxwell's equations and the equations describing the particle dynamics, leads to formidable theoretical and algebraic complexities, especially for warm plasmas. As an effective alternative, the Lagrangian method may be applied. It is shown how this method may be used in the microscopic description of small-signal wave propagation and in the study of nonlinear wave interactions. The linear theory is developed for an infinite, homogeneous, collisionless, warm magnetoplasma. A summary is presented of a perturbation expansion scheme described by Galloway and Kim (1971), and Lagrangians to third order in perturbation are considered. Attention is given to the averaged-Lagrangian density, the action-transfer and coupled-mode equations, and the general solution of the coupled-mode equations.

Rogue waves in the multicomponent Mel'nikov system and multicomponent Schrödinger-Boussinesq system

NASA Astrophysics Data System (ADS)

Sun, Baonan; Lian, Zhan

2018-02-01

By virtue of the bilinear method and the KP hierarchy reduction technique, exact explicit rational solutions of the multicomponent Mel'nikov equation and the multicomponent Schrödinger-Boussinesq equation are constructed, which contain multicomponent short waves and single-component long wave. For the multicomponent Mel'nikov equation, the fundamental rational solutions possess two different behaviours: lump and rogue wave. It is shown that the fundamental (simplest) rogue waves are line localised waves which arise from the constant background with a line profile and then disappear into the constant background again. The fundamental line rogue waves can be classified into three: bright, intermediate and dark line rogue waves. Two subclasses of non-fundamental rogue waves, i.e., multirogue waves and higher-order rogue waves are discussed. The multirogue waves describe interaction of several fundamental line rogue waves, in which interesting wave patterns appear in the intermediate time. Higher-order rogue waves exhibit dynamic behaviours that the wave structures start from lump and then retreat back to it. Moreover, by taking the parameter constraints further, general higher-order rogue wave solutions for the multicomponent Schrödinger-Boussinesq system are generated.

Three-dimensional modelling of thin liquid films over spinning disks

NASA Astrophysics Data System (ADS)

Zhao, Kun; Wray, Alex; Yang, Junfeng; Matar, Omar

2016-11-01

In this research the dynamics of a thin film flowing over a rapidly spinning, horizontal disk is considered. A set of non-axisymmetric evolution equations for the film thickness, radial and azimuthal flow rates are derived using a boundary-layer approximation in conjunction with the Karman-Polhausen approximation for the velocity distribution in the film. These highly nonlinear partial differential equations are then solved numerically in order to reveal the formation of two and three-dimensional large-amplitude waves that travel from the disk inlet to its periphery. The spatio-temporal profile of film thickness provides us with visualization of flow structures over the entire disk and by varying system parameters(volumetric flow rate of fluid and rotational speed of disk) different wave patterns can be observed, including spiral, concentric, smooth waves and wave break-up in exceptional conditions. Similar types of waves can be found by experimentalists in literature and CFD simulation and our results show good agreement with both experimental and CFD results. Furthermore, the semi-parabolic velocity profile assumed in our model under the waves is directly compared with CFD data in various flow regimes in order to validate our model. EPSRC UK Programme Grant EP/K003976/1.

The Hugoniot adiabat of crystalline copper based on molecular dynamics simulation and semiempirical equation of state

NASA Astrophysics Data System (ADS)

Gubin, S. A.; Maklashova, I. V.; Mel'nikov, I. N.

2018-01-01

The molecular dynamics (MD) method was used for prediction of properties of copper under shock-wave compression and clarification of the melting region of crystal copper. The embedded atom potential was used for the interatomic interaction. Parameters of Hugonoit adiabats of solid and liquid phases of copper calculated by the semiempirical Grüneisen equation of state are consistent with the results of MD simulations and experimental data. MD simulation allows to visualize the structure of cooper on the atomistic level. The analysis of the radial distribution function and the standard deviation by MD modeling allows to predict the melting area behind the shock wave front. These MD simulation data are required to verify the wide-range equation of state of metals. The melting parameters of copper based on MD simulations and semiempirical equations of state are consistent with experimental and theoretical data, including the region of the melting point of copper.

Some new traveling wave exact solutions of the (2+1)-dimensional Boiti-Leon-Pempinelli equations.

PubMed

Qi, Jian-ming; Zhang, Fu; Yuan, Wen-jun; Huang, Zi-feng

2014-01-01

We employ the complex method to obtain all meromorphic exact solutions of complex (2+1)-dimensional Boiti-Leon-Pempinelli equations (BLP system of equations). The idea introduced in this paper can be applied to other nonlinear evolution equations. Our results show that all rational and simply periodic traveling wave exact solutions of the equations (BLP) are solitary wave solutions, the complex method is simpler than other methods, and there exist some rational solutions ur,2 (z) and simply periodic solutions us,2-6(z) which are not only new but also not degenerated successively by the elliptic function solutions. We believe that this method should play an important role for finding exact solutions in the mathematical physics. For these new traveling wave solutions, we give some computer simulations to illustrate our main results.

Stability of Viscous St. Venant Roll Waves: From Onset to Infinite Froude Number Limit

NASA Astrophysics Data System (ADS)

Barker, Blake; Johnson, Mathew A.; Noble, Pascal; Rodrigues, L. Miguel; Zumbrun, Kevin

2017-02-01

We study the spectral stability of roll wave solutions of the viscous St. Venant equations modeling inclined shallow water flow, both at onset in the small Froude number or "weakly unstable" limit F→ 2^+ and for general values of the Froude number F, including the limit F→ +∞ . In the former, F→ 2^+, limit, the shallow water equations are formally approximated by a Korteweg-de Vries/Kuramoto-Sivashinsky (KdV-KS) equation that is a singular perturbation of the standard Korteweg-de Vries (KdV) equation modeling horizontal shallow water flow. Our main analytical result is to rigorously validate this formal limit, showing that stability as F→ 2^+ is equivalent to stability of the corresponding KdV-KS waves in the KdV limit. Together with recent results obtained for KdV-KS by Johnson-Noble-Rodrigues-Zumbrun and Barker, this gives not only the first rigorous verification of stability for any single viscous St. Venant roll wave, but a complete classification of stability in the weakly unstable limit. In the remainder of the paper, we investigate numerically and analytically the evolution of the stability diagram as Froude number increases to infinity. Notably, we find transition at around F=2.3 from weakly unstable to different, large- F behavior, with stability determined by simple power-law relations. The latter stability criteria are potentially useful in hydraulic engineering applications, for which typically 2.5≤ F≤ 6.0.

Rogue waves and unbounded solutions of the NLSE

NASA Astrophysics Data System (ADS)

Lechuga, Antonio

2017-04-01

Since the pioneering work of Zakharov has been generally admitted that rogue waves can be studied in the framework of the Nonlinear Schrödinger Equation (NLSE). Many researchers, Akhmediev, Peregrine, Matveev among others gave different solutions to this equation that, in some way, could be linked to rogue waves and also to its more important characteristic: its unexpectedness. Janssen (2003, 2004), Onorato (2004, 2006) and Waseda (2006) linked the coefficient of the nonlinear term of the Schrödinger equation with the Benjamin-Feir index (BFI) that, we know, is a measure of the modulational instability of the waves. From this point of view the value of this coefficient of the NLSE could be known from statistics. Thus the relationship between sea states and the mechanism of generation of rogue waves could be found out. Following the well-known Lie group theory researchers have been studying the Lie point symmetries of the NLSE: the scaling transformations, Galilean transformations and phase transformations. Basically these transformations turn the NLSE into a nonlinear ordinary differential equation called Duffing equation (also called eikonal equation). There are different ways to do this, but in most of them the independent variable that could be seen as a space variable is a kind of moving frame with the time incorporated in this way. The main aim of this work is to classify solutions of the Duffing equation (periodic and nonperiodic waves and also bounded and unbounded waves) bearing in mind that the coefficient of the nonlinear term in the NLSE is left unaltered in the process of the transformation.

Generation of long subharmonic internal waves by surface waves

NASA Astrophysics Data System (ADS)

Tahvildari, Navid; Kaihatu, James M.; Saric, William S.

2016-10-01

A new set of Boussinesq equations is derived to study the nonlinear interactions between long waves in a two-layer fluid. The fluid layers are assumed to be homogeneous, inviscid, incompressible, and immiscible. Based on the Boussinesq equations, an analytical model is developed using a second-order perturbation theory and applied to examine the transient evolution of a resonant triad composed of a surface wave and two oblique subharmonic internal waves. Wave damping due to weak viscosity in both layers is considered. The Boussinesq equations and the analytical model are verified. In contrast to previous studies which focus on short internal waves, we examine long waves and investigate some previously unexplored characteristics of this class of triad interaction. In viscous fluids, surface wave amplitudes must be larger than a threshold to overcome viscous damping and trigger internal waves. The dependency of this critical amplitude as well as the growth and damping rates of internal waves on important parameters in a two-fluid system, namely the directional angle of the internal waves, depth, density, and viscosity ratio of the fluid layers, and surface wave amplitude and frequency is investigated.

On Reductions of the Hirota-Miwa Equation

NASA Astrophysics Data System (ADS)

Hone, Andrew N. W.; Kouloukas, Theodoros E.; Ward, Chloe

2017-07-01

The Hirota-Miwa equation (also known as the discrete KP equation, or the octahedron recurrence) is a bilinear partial difference equation in three independent variables. It is integrable in the sense that it arises as the compatibility condition of a linear system (Lax pair). The Hirota-Miwa equation has infinitely many reductions of plane wave type (including a quadratic exponential gauge transformation), defined by a triple of integers or half-integers, which produce bilinear ordinary difference equations of Somos/Gale-Robinson type. Here it is explained how to obtain Lax pairs and presymplectic structures for these reductions, in order to demonstrate Liouville integrability of some associated maps, certain of which are related to reductions of discrete Toda and discrete KdV equations.

Dark- and bright-rogue-wave solutions for media with long-wave-short-wave resonance.

PubMed

Chen, Shihua; Grelu, Philippe; Soto-Crespo, J M

2014-01-01

Exact explicit rogue-wave solutions of intricate structures are presented for the long-wave-short-wave resonance equation. These vector parametric solutions feature coupled dark- and bright-field counterparts of the Peregrine soliton. Numerical simulations show the robustness of dark and bright rogue waves in spite of the onset of modulational instability. Dark fields originate from the complex interplay between anomalous dispersion and the nonlinearity driven by the coupled long wave. This unusual mechanism, not available in scalar nonlinear wave equation models, can provide a route to the experimental realization of dark rogue waves in, for instance, negative index media or with capillary-gravity waves.

Equivalent equations of motion for gravity and entropy

DOE Office of Scientific and Technical Information (OSTI.GOV)

Czech, Bartlomiej; Lamprou, Lampros; McCandlish, Samuel

We demonstrate an equivalence between the wave equation obeyed by the entanglement entropy of CFT subregions and the linearized bulk Einstein equation in Anti-de Sitter space. In doing so, we make use of the formalism of kinematic space and fields on this space. We show that the gravitational dynamics are equivalent to a gauge invariant wave-equation on kinematic space and that this equation arises in natural correspondence to the conformal Casimir equation in the CFT.

Equivalent equations of motion for gravity and entropy

DOE PAGES

Czech, Bartlomiej; Lamprou, Lampros; McCandlish, Samuel; ...

2017-02-01

We demonstrate an equivalence between the wave equation obeyed by the entanglement entropy of CFT subregions and the linearized bulk Einstein equation in Anti-de Sitter space. In doing so, we make use of the formalism of kinematic space and fields on this space. We show that the gravitational dynamics are equivalent to a gauge invariant wave-equation on kinematic space and that this equation arises in natural correspondence to the conformal Casimir equation in the CFT.

Green's function integral equation method for propagation of electromagnetic waves in an anisotropic dielectric-magnetic slab

NASA Astrophysics Data System (ADS)

Shu, Weixing; Lv, Xiaofang; Luo, Hailu; Wen, Shuangchun

2010-08-01

We extend the Green's function integral method to investigate the propagation of electromagnetic waves through an anisotropic dielectric-magnetic slab. From a microscopic perspective, we analyze the interaction of wave with the slab and derive the propagation characteristics by self-consistent analyses. Applying the results, we find an alternative explanation to the general mechanism for the photon tunneling. The results are confirmed by numerical simulations and disclose the underlying physics of wave propagation through slab. The method extended is applicable to other problems of propagation in dielectric-magnetic materials, including metamaterials.

A convergent series expansion for hyperbolic systems of conservation laws

NASA Technical Reports Server (NTRS)

Harabetian, E.

1985-01-01

The discontinuities piecewise analytic initial value problem for a wide class of conservation laws is considered which includes the full three-dimensional Euler equations. The initial interaction at an arbitrary curved surface is resolved in time by a convergent series. Among other features the solution exhibits shock, contact, and expansion waves as well as sound waves propagating on characteristic surfaces. The expansion waves correspond to he one-dimensional rarefactions but have a more complicated structure. The sound waves are generated in place of zero strength shocks, and they are caused by mismatches in derivatives.

Pile-Driving Pressure and Particle Velocity at the Seabed: Quantifying Effects on Crustaceans and Groundfish.

PubMed

Miller, James H; Potty, Gopu R; Kim, Hui-Kwan

2016-01-01

We modeled the effects of pile driving on crustaceans, groundfish, and other animals near the seafloor. Three different waves were investigated, including the compressional wave, shear wave, and interface wave. A finite element (FE) technique was employed in and around the pile, whereas a parabolic equation (PE) code was used to predict propagation at long ranges from the pile. Pressure, particle displacement, and particle velocity are presented as a function of range at the seafloor for a shallow-water environment near Rhode Island. We discuss the potential effects on animals near the seafloor.

Conservation of wave action. [in discrete oscillating system

NASA Technical Reports Server (NTRS)

Hayes, W. D.

1974-01-01

It is pointed out that two basic principles appear in the theory of wave propagation, including the existence of a phase variable and a law governing the intensity, in terms of a conservation law. The concepts underlying such a conservation law are explored. The waves treated are conservative in the sense that they obey equations derivable from a variational principle applied to a Lagrangian functional. A discrete oscillating system is considered. The approach employed also permits in a natural way the definition of a local action density and flux in problems in which the waves are modal or general.

High Frequency Acoustic Propagation using Level Set Methods

DTIC Science & Technology

2007-01-01

solution of the high frequency approximation to the wave equation. Traditional solutions to the Eikonal equation in high frequency acoustics are...the Eikonal equation derived from the high frequency approximation to the wave equation, ucuH ∇±=∇ )(),( xx , with the nonnegative function c(x...For simplicity, we only consider the case ucuH ∇+=∇ )(),( xx . Two difficulties must be addressed when solving the Eikonal equation in a fixed

Grating formation by a high power radio wave in near-equator ionosphere

DOE Office of Scientific and Technical Information (OSTI.GOV)

Singh, Rohtash; Sharma, A. K.; Tripathi, V. K.

2011-11-15

The formation of a volume grating in the near-equator regions of ionosphere due to a high power radio wave is investigated. The radio wave, launched from a ground based transmitter, forms a standing wave pattern below the critical layer, heating the electrons in a space periodic manner. The thermal conduction along the magnetic lines of force inhibits the rise in electron temperature, limiting the efficacy of heating to within a latitude of few degrees around the equator. The space periodic electron partial pressure leads to ambipolar diffusion creating a space periodic density ripple with wave vector along the vertical. Suchmore » a volume grating is effective to cause strong reflection of radio waves at a frequency one order of magnitude higher than the maximum plasma frequency in the ionosphere. Linearly mode converted plasma wave could scatter even higher frequency radio waves.« less

Stability and Metastability of Trapless Bose-Einstein Condensates and Quantum Liquids

NASA Astrophysics Data System (ADS)

Zloshchastiev, Konstantin G.

2017-07-01

Various kinds of Bose-Einstein condensates are considered, which evolve without any geometric constraints or external trap potentials including gravitational. For studies of their collective oscillations and stability, including the metastability and macroscopic tunneling phenomena, both the variational approach and the Vakhitov-Kolokolov (VK) criterion are employed; calculations are done for condensates of an arbitrary spatial dimension. It is determined that that the trapless condensate described by the logarithmic wave equation is essentially stable, regardless of its dimensionality, while the trapless condensates described by wave equations of a polynomial type with respect to the wavefunction, such as the Gross-Pitaevskii (cubic), cubic-quintic, and so on, are at best metastable. This means that trapless "polynomial" condensates are unstable against spontaneous delocalization caused by fluctuations of their width, density and energy, leading to a finite lifetime.

Exact Solutions of Atmospheric (2+1)-Dimensional Nonlinear Incompressible Non-hydrostatic Boussinesq Equations

NASA Astrophysics Data System (ADS)

Liu, Ping; Wang, Ya-Xiong; Ren, Bo; Li, Jin-Hua

2016-12-01

Exact solutions of the atmospheric (2+1)-dimensional nonlinear incompressible non-hydrostatic Boussinesq (INHB) equations are researched by Combining function expansion and symmetry method. By function expansion, several expansion coefficient equations are derived. Symmetries and similarity solutions are researched in order to obtain exact solutions of the INHB equations. Three types of symmetry reduction equations and similarity solutions for the expansion coefficient equations are proposed. Non-traveling wave solutions for the INHB equations are obtained by symmetries of the expansion coefficient equations. Making traveling wave transformations on expansion coefficient equations, we demonstrate some traveling wave solutions of the INHB equations. The evolutions on the wind velocities, temperature perturbation and pressure perturbation are demonstrated by figures, which demonstrate the periodic evolutions with time and space. Supported by the National Natural Science Foundation of China under Grant Nos. 11305031 and 11305106, and Training Programme Foundation for Outstanding Young Teachers in Higher Education Institutions of Guangdong Province under Grant No. Yq2013205

Applications of the ETEM for obtaining optical soliton solutions for the Lakshmanan-Porsezian-Daniel model

NASA Astrophysics Data System (ADS)

Manafian, Jalil; Foroutan, Mohammadreza; Guzali, Aref

2017-11-01

This paper examines the effectiveness of an integration scheme which is called the extended trial equation method (ETEM) for solving a well-known nonlinear equation of partial differential equations (PDEs). In this respect, the Lakshmanan-Porsezian-Daniel (LPD) equation with Kerr and power laws of nonlinearity which describes higher-order dispersion, full nonlinearity and spatiotemporal dispersion is considered, and as an achievement, a series of exact travelling-wave solutions for the aforementioned equation is formally extracted. Explicit new exact solutions are derived in different form such as dark solitons, bright solitons, solitary wave, periodic solitary wave, rational function, and elliptic function solutions of LPD equation. The movement of obtained solutions is shown graphically, which helps to understand the physical phenomena of this optical soliton equation. Many other such types of nonlinear equations arising in basic fabric of communications network technology and nonlinear optics can also be solved by this method.

Time domain viscoelastic full waveform inversion

NASA Astrophysics Data System (ADS)

Fabien-Ouellet, Gabriel; Gloaguen, Erwan; Giroux, Bernard

2017-06-01

Viscous attenuation can have a strong impact on seismic wave propagation, but it is rarely taken into account in full waveform inversion (FWI). When viscoelasticity is considered in time domain FWI, the displacement formulation of the wave equation is usually used instead of the popular velocity-stress formulation. However, inversion schemes rely on the adjoint equations, which are quite different for the velocity-stress formulation than for the displacement formulation. In this paper, we apply the adjoint state method to the isotropic viscoelastic wave equation in the velocity-stress formulation based on the generalized standard linear solid rheology. By applying linear transformations to the wave equation before deriving the adjoint state equations, we obtain two symmetric sets of partial differential equations for the forward and adjoint variables. The resulting sets of equations only differ by a sign change and can be solved by the same numerical implementation. We also investigate the crosstalk between parameter classes (velocity and attenuation) of the viscoelastic equation. More specifically, we show that the attenuation levels can be used to recover the quality factors of P and S waves, but that they are very sensitive to velocity errors. Finally, we present a synthetic example of viscoelastic FWI in the context of monitoring CO2 geological sequestration. We show that FWI based on our formulation can indeed recover P- and S-wave velocities and their attenuation levels when attenuation is high enough. Both changes in velocity and attenuation levels recovered with FWI can be used to track the CO2 plume during and after injection. Further studies are required to evaluate the performance of viscoelastic FWI on real data.

Semi-analytical Karhunen-Loeve representation of irregular waves based on the prolate spheroidal wave functions

NASA Astrophysics Data System (ADS)

Lee, Gibbeum; Cho, Yeunwoo

2018-01-01

A new semi-analytical approach is presented to solving the matrix eigenvalue problem or the integral equation in Karhunen-Loeve (K-L) representation of random data such as irregular ocean waves. Instead of direct numerical approach to this matrix eigenvalue problem, which may suffer from the computational inaccuracy for big data, a pair of integral and differential equations are considered, which are related to the so-called prolate spheroidal wave functions (PSWF). First, the PSWF is expressed as a summation of a small number of the analytical Legendre functions. After substituting them into the PSWF differential equation, a much smaller size matrix eigenvalue problem is obtained than the direct numerical K-L matrix eigenvalue problem. By solving this with a minimal numerical effort, the PSWF and the associated eigenvalue of the PSWF differential equation are obtained. Then, the eigenvalue of the PSWF integral equation is analytically expressed by the functional values of the PSWF and the eigenvalues obtained in the PSWF differential equation. Finally, the analytically expressed PSWFs and the eigenvalues in the PWSF integral equation are used to form the kernel matrix in the K-L integral equation for the representation of exemplary wave data such as ordinary irregular waves. It is found that, with the same accuracy, the required memory size of the present method is smaller than that of the direct numerical K-L representation and the computation time of the present method is shorter than that of the semi-analytical method based on the sinusoidal functions.

Low-cost ultrasonic lamb-wave transducer

NASA Technical Reports Server (NTRS)

Kammerer, C. C.

1978-01-01

Transducer propagates Lamb wave through thin aluminum sheet material. Model includes two elements that measure effects of damping and loading which, in turn, are indirectly equated to bond integrity. Transducer has been used to evaluate bond integrity of aluminum facing adhesively bonded to aluminum facing. Because of versatility, it is now possible to inspect many objects of different configurations that could not be reached with earlier transducers.

Shock waves: The Maxwell-Cattaneo case.

PubMed

Uribe, F J

2016-03-01

Several continuum theories for shock waves give rise to a set of differential equations in which the analysis of the underlying vector field can be done using the tools of the theory of dynamical systems. We illustrate the importance of the divergences associated with the vector field by considering the ideas by Maxwell and Cattaneo and apply them to study shock waves in dilute gases. By comparing the predictions of the Maxwell-Cattaneo equations with shock wave experiments we are lead to the following conclusions: (a) For low compressions (low Mach numbers: M) the results from the Maxwell-Cattaneo equations provide profiles that are in fair agreement with the experiments, (b) as the Mach number is increased we find a range of Mach numbers (1.27 ≈ M(1) < M < M(2) ≈ 1.90) such that numerical shock wave solutions to the Maxwell-Cattaneo equations cannot be found, and (c) for greater Mach numbers (M>M_{2}) shock wave solutions can be found though they differ significantly from experiments.

Lagrangian geometrical optics of nonadiabatic vector waves and spin particles

DOE PAGES

Ruiz, D. E.; Dodin, I. Y.

2015-07-29

Linear vector waves, both quantum and classical, experience polarization-driven bending of ray trajectories and polarization dynamics that can be interpreted as the precession of the "wave spin". Here, both phenomena are governed by an effective gauge Hamiltonian vanishing in leading-order geometrical optics. This gauge Hamiltonian can be recognized as a generalization of the Stern-Gerlach Hamiltonian that is commonly known for spin-1/2 quantum particles. The corresponding reduced Lagrangians for continuous nondissipative waves and their geometrical-optics rays are derived from the fundamental wave Lagrangian. The resulting Euler-Lagrange equations can describe simultaneous interactions of N resonant modes, where N is arbitrary, and leadmore » to equations for the wave spin, which happens to be an (N 2 - 1)-dimensional spin vector. As a special case, classical equations for a Dirac particle (N = 2) are deduced formally, without introducing additional postulates or interpretations, from the Dirac quantum Lagrangian with the Pauli term. The model reproduces the Bargmann-Michel-Telegdi equations with added Stern-Gerlach force.« less

4-wave dynamics in kinetic wave turbulence

NASA Astrophysics Data System (ADS)

Chibbaro, Sergio; Dematteis, Giovanni; Rondoni, Lamberto

2018-01-01

A general Hamiltonian wave system with quartic resonances is considered, in the standard kinetic limit of a continuum of weakly interacting dispersive waves with random phases. The evolution equation for the multimode characteristic function Z is obtained within an ;interaction representation; and a perturbation expansion in the small nonlinearity parameter. A frequency renormalization is performed to remove linear terms that do not appear in the 3-wave case. Feynman-Wyld diagrams are used to average over phases, leading to a first order differential evolution equation for Z. A hierarchy of equations, analogous to the Boltzmann hierarchy for low density gases is derived, which preserves in time the property of random phases and amplitudes. This amounts to a general formalism for both the N-mode and the 1-mode PDF equations for 4-wave turbulent systems, suitable for numerical simulations and for investigating intermittency. Some of the main results which are developed here in detail have been tested numerically in a recent work.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Li Jibin; Qiao Zhijun

This paper deals with the following equation m{sub t}=(1/2)(1/m{sup k}){sub xxx}-(1/2)(1/m{sup k}){sub x}, which is proposed by Z. J. Qiao [J. Math. Phys. 48, 082701 (2007)] and Qiao and Liu [Chaos, Solitons Fractals 41, 587 (2009)]. By adopting the phase analysis method of planar dynamical systems and the theory of the singular traveling wave systems to the traveling wave solutions of the equation, it is shown that for different k, the equation may have infinitely many solitary wave solutions, periodic wave solutions, kink/antikink wave solutions, cusped solitary wave solutions, and breaking loop solutions. We discuss in a detail the casesmore » of k=-2,-(1/2),(1/2),2, and parametric representations of all possible bounded traveling wave solutions are given in the different (c,g)-parameter regions.« less

Low-frequency surface waves on semi-bounded magnetized quantum plasma

DOE Office of Scientific and Technical Information (OSTI.GOV)

Moradi, Afshin, E-mail: a.moradi@kut.ac.ir

2016-08-15

The propagation of low-frequency electrostatic surface waves on the interface between a vacuum and an electron-ion quantum plasma is studied in the direction perpendicular to an external static magnetic field which is parallel to the interface. A new dispersion equation is derived by employing both the quantum magnetohydrodynamic and Poisson equations. It is shown that the dispersion equations for forward and backward-going surface waves are different from each other.

Calculation Of Pneumatic Attenuation In Pressure Sensors

NASA Technical Reports Server (NTRS)

Whitmore, Stephen A.

1991-01-01

Errors caused by attenuation of air-pressure waves in narrow tubes calculated by method based on fundamental equations of flow. Changes in ambient pressure transmitted along narrow tube to sensor. Attenuation of high-frequency components of pressure wave calculated from wave equation derived from Navier-Stokes equations of viscous flow in tube. Developed to understand and compensate for frictional attenuation in narrow tubes used to connect aircraft pressure sensors with pressure taps on affected surfaces.

Electromagnetic or other directed energy pulse launcher

DOEpatents

Ziolkowski, Richard W.

1990-01-01

The physical realization of new solutions of wave propagation equations, such as Maxwell's equations and the scaler wave equation, produces localized pulses of wave energy such as electromagnetic or acoustic energy which propagate over long distances without divergence. The pulses are produced by driving each element of an array of radiating sources with a particular drive function so that the resultant localized packet of energy closely approximates the exact solutions and behaves the same.

Traveling wave solutions of the Boussinesq equation via the new approach of generalized (G'/G)-expansion method.

PubMed

Alam, Md Nur; Akbar, M Ali; Roshid, Harun-Or-

2014-01-01

Exact solutions of nonlinear evolution equations (NLEEs) play a vital role to reveal the internal mechanism of complex physical phenomena. In this work, the exact traveling wave solutions of the Boussinesq equation is studied by using the new generalized (G'/G)-expansion method. Abundant traveling wave solutions with arbitrary parameters are successfully obtained by this method and the wave solutions are expressed in terms of the hyperbolic, trigonometric, and rational functions. It is shown that the new approach of generalized (G'/G)-expansion method is a powerful and concise mathematical tool for solving nonlinear partial differential equations in mathematical physics and engineering. 05.45.Yv, 02.30.Jr, 02.30.Ik.

FAST TRACK COMMUNICATION: Soliton solutions of the KP equation with V-shape initial waves

NASA Astrophysics Data System (ADS)

Kodama, Y.; Oikawa, M.; Tsuji, H.

2009-08-01

We consider the initial value problems of the Kadomtsev-Petviashvili (KP) equation for symmetric V-shape initial waves consisting of two semi-infinite line solitons with the same amplitude. Those are particularly important for studies of large amplitude waves such as tsunami in shallow water. Numerical simulations show that the solutions of the initial value problem approach asymptotically to certain exact solutions of the KP equation found recently in [1]. We then use a chord diagram to explain the asymptotic result. This provides an analytical method to study asymptotic behavior for the initial value problem of the KP equation. We also demonstrate a real experiment of shallow water waves which may represent the solution discussed in this communication.

Dynamic behavior of ion acoustic waves in electron-positron-ion magnetoplasmas with superthermal electrons and positrons

DOE Office of Scientific and Technical Information (OSTI.GOV)

Saha, Asit, E-mail: asit-saha123@rediffmail.com, E-mail: prasantachatterjee1@rediffmail.com; Department of Mathematics, Siksha Bhavana, Visva Bharati University, Santiniketan-731235; Pal, Nikhil

The dynamic behavior of ion acoustic waves in electron-positron-ion magnetoplasmas with superthermal electrons and positrons has been investigated in the framework of perturbed and non-perturbed Kadomtsev-Petviashili (KP) equations. Applying the reductive perturbation technique, we have derived the KP equation in electron-positron-ion magnetoplasma with kappa distributed electrons and positrons. Bifurcations of ion acoustic traveling waves of the KP equation are presented. Using the bifurcation theory of planar dynamical systems, the existence of the solitary wave solutions and the periodic traveling wave solutions has been established. Two exact solutions of these waves have been derived depending on the system parameters. Then, usingmore » the Hirota's direct method, we have obtained two-soliton and three-soliton solutions of the KP equation. The effect of the spectral index κ on propagations of the two-soliton and the three-soliton has been shown. Considering an external periodic perturbation, we have presented the quasi periodic behavior of ion acoustic waves in electron-positron-ion magnetoplasmas.« less

Stable two-dimensional solitary pulses in linearly coupled dissipative Kadomtsev-Petviashvili equations.

PubMed

Feng, Bao-Feng; Malomed, Boris A; Kawahara, Takuji

2002-11-01

We present a two-dimensional (2D) generalization of the stabilized Kuramoto-Sivashinsky system, based on the Kadomtsev-Petviashvili (KP) equation including dissipation of the generic [Newell-Whitehead-Segel (NWS)] type and gain. The system directly applies to the description of gravity-capillary waves on the surface of a liquid layer flowing down an inclined plane, with a surfactant diffusing along the layer's surface. Actually, the model is quite general, offering a simple way to stabilize nonlinear media, combining the weakly 2D dispersion of the KP type with gain and NWS dissipation. Other applications are internal waves in multilayer fluids flowing down an inclined plane, double-front flames in gaseous mixtures, etc. Parallel to this weakly 2D model, we also introduce and study a semiphenomenological one, whose dissipative terms are isotropic, rather than of the NWS type, in order to check if qualitative results are sensitive to the exact form of the lossy terms. The models include an additional linear equation of the advection-diffusion type, linearly coupled to the main KP-NWS equation. The extra equation provides for stability of the zero background in the system, thus opening a way for the existence of stable localized pulses. We focus on the most interesting case, when the dispersive part of the system is of the KP-I type, which corresponds, e.g., to capillary waves, and makes the existence of completely localized 2D pulses possible. Treating the losses and gain as small perturbations and making use of the balance equation for the field momentum, we find that the equilibrium between the gain and losses may select two steady-state solitons from their continuous family existing in the absence of the dissipative terms (the latter family is found in an exact analytical form, and is numerically demonstrated to be stable). The selected soliton with the larger amplitude is expected to be stable. Direct simulations completely corroborate the analytical predictions, for both the physical and phenomenological models.

Complex Riccati equations as a link between different approaches for the description of dissipative and irreversible systems

NASA Astrophysics Data System (ADS)

Schuch, Dieter

2012-08-01

Quantum mechanics is essentially described in terms of complex quantities like wave functions. The interesting point is that phase and amplitude of the complex wave function are not independent of each other, but coupled by some kind of conservation law. This coupling exists in time-independent quantum mechanics and has a counterpart in its time-dependent form. It can be traced back to a reformulation of quantum mechanics in terms of nonlinear real Ermakov equations or equivalent complex nonlinear Riccati equations, where the quadratic term in the latter equation explains the origin of the phase-amplitude coupling. Since realistic physical systems are always in contact with some kind of environment this aspect is also taken into account. In this context, different approaches for describing open quantum systems, particularly effective ones, are discussed and compared. Certain kinds of nonlinear modifications of the Schrödinger equation are discussed as well as their interrelations and their relations to linear approaches via non-unitary transformations. The modifications of the aforementioned Ermakov and Riccati equations when environmental effects are included can be determined in the time-dependent case. From formal similarities conclusions can be drawn how the equations of time-independent quantum mechanics can be modified to also incluce the enviromental aspects.

Extremely Fast Numerical Integration of Ocean Surface Wave Dynamics: Building Blocks for a Higher Order Method

DTIC Science & Technology

2006-09-30

equation known as the Kadomtsev - Petviashvili (KP) equation ): (ηt + coηx +αηηx + βη )x +γηyy = 0 (4) where γ = co / 2 . The KdV equation ...using the spectral formulation of the Kadomtsev - Petviashvili equation , a standard equation for nonlinear, shallow water wave dynamics that is a... Petviashvili and nonlinear Schroedinger equations and higher order corrections have been developed as prerequisites to coding the Boussinesq and Euler

Multiphase wavetrains, singular wave interactions and the emergence of the Korteweg–de Vries equation

PubMed Central

Bridges, Thomas J.

2016-01-01

Multiphase wavetrains are multiperiodic travelling waves with a set of distinct wavenumbers and distinct frequencies. In conservative systems, such families are associated with the conservation of wave action or other conservation law. At generic points (where the Jacobian of the wave action flux is non-degenerate), modulation of the wavetrain leads to the dispersionless multiphase conservation of wave action. The main result of this paper is that modulation of the multiphase wavetrain, when the Jacobian of the wave action flux vector is singular, morphs the vector-valued conservation law into the scalar Korteweg–de Vries (KdV) equation. The coefficients in the emergent KdV equation have a geometrical interpretation in terms of projection of the vector components of the conservation law. The theory herein is restricted to two phases to simplify presentation, with extensions to any finite dimension discussed in the concluding remarks. Two applications of the theory are presented: a coupled nonlinear Schrödinger equation and two-layer shallow-water hydrodynamics with a free surface. Both have two-phase solutions where criticality and the properties of the emergent KdV equation can be determined analytically. PMID:28119546

Chirped solitary pulses for a nonic nonlinear Schrödinger equation on a continuous-wave background

NASA Astrophysics Data System (ADS)

Triki, Houria; Porsezian, K.; Choudhuri, Amitava; Dinda, P. Tchofo

2016-06-01

A class of derivative nonlinear Schrödinger equation with cubic-quintic-septic-nonic nonlinear terms describing the propagation of ultrashort optical pulses through a nonlinear medium with higher-order Kerr responses is investigated. An intensity-dependent chirp ansatz is adopted for solving the two coupled amplitude-phase nonlinear equations of the propagating wave. We find that the dynamics of field amplitude in this system is governed by a first-order nonlinear ordinary differential equation with a tenth-degree nonlinear term. We demonstrate that this system allows the propagation of a very rich variety of solitary waves (kink, dark, bright, and gray solitary pulses) which do not coexist in the conventional nonlinear systems that have appeared so far in the literature. The stability of the solitary wave solution under some violation on the parametric conditions is investigated. Moreover, we show that, unlike conventional systems, the nonlinear Schrödinger equation considered here meets the special requirements for the propagation of a chirped solitary wave on a continuous-wave background, involving a balance among group velocity dispersion, self-steepening, and higher-order nonlinearities of different nature.

Comparison between results of solution of Burgers' equation and Laplace's equation by Galerkin and least-square finite element methods

NASA Astrophysics Data System (ADS)

Adib, Arash; Poorveis, Davood; Mehraban, Farid

2018-03-01

In this research, two equations are considered as examples of hyperbolic and elliptic equations. In addition, two finite element methods are applied for solving of these equations. The purpose of this research is the selection of suitable method for solving each of two equations. Burgers' equation is a hyperbolic equation. This equation is a pure advection (without diffusion) equation. This equation is one-dimensional and unsteady. A sudden shock wave is introduced to the model. This wave moves without deformation. In addition, Laplace's equation is an elliptical equation. This equation is steady and two-dimensional. The solution of Laplace's equation in an earth dam is considered. By solution of Laplace's equation, head pressure and the value of seepage in the directions X and Y are calculated in different points of earth dam. At the end, water table is shown in the earth dam. For Burgers' equation, least-square method can show movement of wave with oscillation but Galerkin method can not show it correctly (the best method for solving of the Burgers' equation is discrete space by least-square finite element method and discrete time by forward difference.). For Laplace's equation, Galerkin and least square methods can show water table correctly in earth dam.

Numerical studies of the KP line-solitons

NASA Astrophysics Data System (ADS)

Chakravarty, S.; McDowell, T.; Osborne, M.

2017-03-01

The Kadomtsev-Petviashvili (KP) equation admits a class of solitary wave solutions localized along distinct rays in the xy-plane, called the line-solitons, which describe the interaction of shallow water waves on a flat surface. These wave interactions have been observed on long, flat beaches, as well as have been recreated in laboratory experiments. In this paper, the line-solitons are investigated via direct numerical simulations of the KP equation, and the interactions of the evolved solitary wave patterns are studied. The objective is to obtain greater insight into solitary wave interactions in shallow water and to determine the extent the KP equation is a good model in describing these nonlinear interactions.

Two-dimensional cylindrical ion-acoustic solitary and rogue waves in ultrarelativistic plasmas

DOE Office of Scientific and Technical Information (OSTI.GOV)

Ata-ur-Rahman; National Centre for Physics at QAU Campus, Shahdrah Valley Road, Islamabad 44000; Ali, S.

2013-07-15

The propagation of ion-acoustic (IA) solitary and rogue waves is investigated in a two-dimensional ultrarelativistic degenerate warm dense plasma. By using the reductive perturbation technique, the cylindrical Kadomtsev–Petviashvili (KP) equation is derived, which can be further transformed into a Korteweg–de Vries (KdV) equation. The latter admits a solitary wave solution. However, when the frequency of the carrier wave is much smaller than the ion plasma frequency, the KdV equation can be transferred to a nonlinear Schrödinger equation to study the nonlinear evolution of modulationally unstable modified IA wavepackets. The propagation characteristics of the IA solitary and rogue waves are stronglymore » influenced by the variation of different plasma parameters in an ultrarelativistic degenerate dense plasma. The present results might be helpful to understand the nonlinear electrostatic excitations in astrophysical degenerate dense plasmas.« less

A Self-Consistent Model of the Interacting Ring Current Ions with Electromagnetic ICWs

NASA Technical Reports Server (NTRS)

Khazanov, G. V.; Gamayunov, K. V.; Jordanova, V. K.; Krivorutsky, E. N.; Whitaker, Ann F. (Technical Monitor)

2001-01-01

Initial results from a newly developed model of the interacting ring current ions and ion cyclotron waves are presented. The model is based on the system of two bound kinetic equations: one equation describes the ring current ion dynamics, and another equation describes wave evolution. The system gives a self-consistent description of ring current ions and ion cyclotron waves in a quasilinear approach. These two equations were solved on a global scale under non steady-state conditions during the May 2-5, 1998 storm. The structure and dynamics of the ring current proton precipitating flux regions and the wave active zones at three time cuts around initial, main, and late recovery phases of the May 4, 1998 storm phase are presented and discussed in detail. Comparisons of the model wave-ion data with the Polar/HYDRA and Polar/MFE instruments results are presented..

Equations for description of nonlinear standing waves in constant-cross-sectioned resonators.

PubMed

Bednarik, Michal; Cervenka, Milan

2014-03-01

This work is focused on investigation of applicability of two widely used model equations for description of nonlinear standing waves in constant-cross-sectioned resonators. The investigation is based on the comparison of numerical solutions of these model equations with solutions of more accurate model equations whose validity has been verified experimentally in a number of published papers.

Sound Beams with Shockwave Pulses

NASA Astrophysics Data System (ADS)

Enflo, B. O.

2000-11-01

The beam equation for a sound beam in a diffusive medium, called the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation, has a class of solutions, which are power series in the transverse variable with the terms given by a solution of a generalized Burgers’ equation. A free parameter in this generalized Burgers’ equation can be chosen so that the equation describes an N-wave which does not decay. If the beam source has the form of a spherical cap, then a beam with a preserved shock can be prepared. This is done by satisfying an inequality containing the spherical radius, the N-wave pulse duration, the N-wave pulse amplitude, and the sound velocity in the fluid.

A Three-Wave Model of the Stratosphere with Coupled Dynamics, Radiation and Photochemistry. Appendix M

NASA Technical Reports Server (NTRS)

Shia, Run-Lie; Zhou, Shuntai; Ko, Malcolm K. W.; Sze, Nien-Dak; Salstein, David; Cady-Pereira, Karen

1997-01-01

A zonal mean chemistry transport model (2-D CTM) coupled with a semi-spectral dynamical model is used to simulate the distributions of trace gases in the present day atmosphere. The zonal-mean and eddy equations for the velocity and the geopotential height are solved in the semi-spectral dynamical model. The residual mean circulation is derived from these dynamical variables and used to advect the chemical species in the 2- D CTM. Based on a linearized wave transport equation, the eddy diffusion coefficients for chemical tracers are expressed in terms of the amplitude, frequency and growth rate of dynamical waves; local chemical loss rates; and a time constant parameterizing small scale mixing. The contributions to eddy flux are from the time varying wave amplitude (transient eddy), chemical reactions (chemical eddy) and small scale mixing. In spite of the high truncation in the dynamical module (only three longest waves are resolved), the model has simulated many observed characteristics of stratospheric dynamics and distribution of chemical species including ozone. Compared with the values commonly used in 2-D CTMs, the eddy diffusion coefficients for chemical species calculated in this model are smaller, especially in the subtropics. It is also found that the chemical eddy diffusion has only a small effects in determining the distribution of most slow species, including ozone in the stratosphere.

Wave-Kinetic Simulations of the Nonlinear Generation of Electromagnetic VLF Waves through Velocity Ring Instabilities

NASA Astrophysics Data System (ADS)

Ganguli, G.; Crabtree, C. E.; Rudakov, L.; Mithaiwala, M.

2014-12-01

Velocity ring instabilities are a common naturally occuring magnetospheric phenomenon that can also be generated by man made ionospheric experiments. These instabilities are known to generate lower-hybrid waves, which generally cannot propagte out of the source region. However, nonlinear wave physics can convert these linearly driven electrostatic lower-hybrid waves into electromagnetic waves that can escape the source region. These nonlinearly generated waves can be an important source of VLF turbulence that controls the trapped electron lifetime in the radiation belts. We develop numerical solutions to the wave-kinetic equation in a periodic box including the effects of nonlinear (NL) scattering (nonlinear Landau damping) of Lower-hybrid waves giving the evolution of the wave-spectra in wavenumber space. Simultaneously we solve the particle diffusion equation of both the background plasma particles and the ring ions, due to both linear and nonlinear Landau resonances. At initial times for cold ring ions, an electrostatic beam mode is excited, while the kinetic mode is stable. As the instability progresses the ring ions heat, the beam mode is stabilized, and the kinetic mode destabilizes. When the amplitude of the waves becomes sufficient the lower-hybrid waves are scattered (by either nearly unmagnetized ions or magnetized electrons) into electromagnetic magnetosonic waves [Ganguli et al 2010]. The effect of NL scattering is to limit the amplitude of the waves, slowing down the quasilinear relaxation time and ultimately allowing more energy from the ring to be liberated into waves [Mithaiwala et al. 2011]. The effects of convection out of the instability region are modeled, additionally limiting the amplitude of the waves, allowing further energy to be liberated from the ring [Scales et al., 2012]. Results are compared to recent 3D PIC simulations [Winske and Duaghton 2012].

Conservative, unconditionally stable discretization methods for Hamiltonian equations, applied to wave motion in lattice equations modeling protein molecules

NASA Astrophysics Data System (ADS)

LeMesurier, Brenton

2012-01-01

A new approach is described for generating exactly energy-momentum conserving time discretizations for a wide class of Hamiltonian systems of DEs with quadratic momenta, including mechanical systems with central forces; it is well-suited in particular to the large systems that arise in both spatial discretizations of nonlinear wave equations and lattice equations such as the Davydov System modeling energetic pulse propagation in protein molecules. The method is unconditionally stable, making it well-suited to equations of broadly “Discrete NLS form”, including many arising in nonlinear optics. Key features of the resulting discretizations are exact conservation of both the Hamiltonian and quadratic conserved quantities related to continuous linear symmetries, preservation of time reversal symmetry, unconditional stability, and respecting the linearity of certain terms. The last feature allows a simple, efficient iterative solution of the resulting nonlinear algebraic systems that retain unconditional stability, avoiding the need for full Newton-type solvers. One distinction from earlier work on conservative discretizations is a new and more straightforward nearly canonical procedure for constructing the discretizations, based on a “discrete gradient calculus with product rule” that mimics the essential properties of partial derivatives. This numerical method is then used to study the Davydov system, revealing that previously conjectured continuum limit approximations by NLS do not hold, but that sech-like pulses related to NLS solitons can nevertheless sometimes arise.

Lump Solitons in Surface Tension Dominated Flows

NASA Astrophysics Data System (ADS)

Milewski, Paul; Berger, Kurt

1999-11-01

The Kadomtsev-Petviashvilli I equation (KPI) which models small-amplitude, weakly three-dimensional surface-tension dominated long waves is integrable and allows for algebraically decaying lump solitary waves. It is not known (theoretically or numerically) whether the full free-surface Euler equations support such solutions. We consider an intermediate model, the generalised Benney-Luke equation (gBL) which is isotropic (not weakly three-dimensional) and contains KPI as a limit. We show numerically that: 1. gBL supports lump solitary waves; 2. These waves collide elastically and are stable; 3. They are generated by resonant flow over an obstacle.

Shock wave equation of state of muscovite

NASA Technical Reports Server (NTRS)

Sekine, Toshimori; Rubin, Allan M.; Ahrens, Thomas J.

1991-01-01

Shock wave data were obtained between 20 and 140 GPa for natural muscovite obtained from Methuen Township (Ontario), in order to provide a shock-wave equation of state for this crustal hydrous mineral. The shock equation of state data could be fit by a linear shock velocity (Us) versus particle velocity (Up) relation Us = 4.62 + 1.27 Up (km/s). Third-order Birch-Murnaghan equation of state parameters were found to be K(OS) = 52 +/-4 GPa and K-prime(OS) = 3.2 +/-0.3 GPa. These parameters are comparable to those of other hydrous minerals such as brucite, serpentine, and tremolite.

Two-dimensional interaction of a shear flow with a free surface in a stratified fluid and its solitary-wave solutions via mathematical methods

NASA Astrophysics Data System (ADS)

Seadawy, Aly R.

2017-12-01

In this study, we presented the problem formulations of models for internal solitary waves in a stratified shear flow with a free surface. The nonlinear higher order of extended KdV equations for the free surface displacement is generated. We derived the coefficients of the nonlinear higher-order extended KdV equation in terms of integrals of the modal function for the linear long-wave theory. The wave amplitude potential and the fluid pressure of the extended KdV equation in the form of solitary-wave solutions are deduced. We discussed and analyzed the stability of the obtained solutions and the movement role of the waves by making graphs of the exact solutions.

Nonlinear interactions between electromagnetic waves and electron plasma oscillations in quantum plasmas.

PubMed

Shukla, P K; Eliasson, B

2007-08-31

We consider nonlinear interactions between intense circularly polarized electromagnetic (CPEM) waves and electron plasma oscillations (EPOs) in a dense quantum plasma, taking into account the electron density response in the presence of the relativistic ponderomotive force and mass increase in the CPEM wave fields. The dynamics of the CPEM waves and EPOs is governed by the two coupled nonlinear Schrödinger equations and Poisson's equation. The nonlinear equations admit the modulational instability of an intense CPEM pump wave against EPOs, leading to the formation and trapping of localized CPEM wave pipes in the electron density hole that is associated with a positive potential distribution in our dense plasma. The relevance of our investigation to the next generation intense laser-solid density plasma interaction experiments is discussed.

Nonlinear propagation of light in Dirac matter.

PubMed

Eliasson, Bengt; Shukla, P K

2011-09-01

The nonlinear interaction between intense laser light and a quantum plasma is modeled by a collective Dirac equation coupled with the Maxwell equations. The model is used to study the nonlinear propagation of relativistically intense laser light in a quantum plasma including the electron spin-1/2 effect. The relativistic effects due to the high-intensity laser light lead, in general, to a downshift of the laser frequency, similar to a classical plasma where the relativistic mass increase leads to self-induced transparency of laser light and other associated effects. The electron spin-1/2 effects lead to a frequency upshift or downshift of the electromagnetic (EM) wave, depending on the spin state of the plasma and the polarization of the EM wave. For laboratory solid density plasmas, the spin-1/2 effects on the propagation of light are small, but they may be significant in superdense plasma in the core of white dwarf stars. We also discuss extensions of the model to include kinetic effects of a distribution of the electrons on the nonlinear propagation of EM waves in a quantum plasma.

Relativistic shock waves in an electron-positron plasma

NASA Astrophysics Data System (ADS)

Tsintsadze, Levan N.

1995-12-01

The equations describing the detailed structure of radiation electromagnetic hydrodynamics for a relativistically hot electron-positron plasma are derived. Various discontinuities are studied by these equations. It is shown that the dependence of the electron (positron) mass on the temperature changes the structure of discontinuities, including shock waves, both qualitatively and quantitatively. Steady radiative shocks are considered, which can arise in steady flows, and which also can be used to describe the propagation of shocks when the shock thickness is very small as compared to the characteristic length over which the ambient medium changes significantly. First, the magnetohydrodynamic shock wave is treated as a discontinuity and jump relations, which relate the equilibrium states of the upstream and downstream plasma far from the front, are derived. Then the structure of the front itself is considered and tangential, contact (or entropy) and rotational discontinuities are investigated.

Numerical Study of Hydrothermal Wave Suppression in Thermocapillary Flow Using a Predictive Control Method

NASA Astrophysics Data System (ADS)

Muldoon, F. H.

2018-04-01

Hydrothermal waves in flows driven by thermocapillary and buoyancy effects are suppressed by applying a predictive control method. Hydrothermal waves arise in the manufacturing of crystals, including the "open boat" crystal growth process, and lead to undesirable impurities in crystals. The open boat process is modeled using the two-dimensional unsteady incompressible Navier-Stokes equations under the Boussinesq approximation and the linear approximation of the surface thermocapillary force. The flow is controlled by a spatially and temporally varying heat flux density through the free surface. The heat flux density is determined by a conjugate gradient optimization algorithm. The gradient of the objective function with respect to the heat flux density is found by solving adjoint equations derived from the Navier-Stokes ones in the Boussinesq approximation. Special attention is given to heat flux density distributions over small free-surface areas and to the maximum admissible heat flux density.

Achieving accuracy in first-principles calculations at extreme temperature and pressure

NASA Astrophysics Data System (ADS)

Mattsson, Ann; Wills, John

2013-06-01

First-principles calculations are increasingly used to provide EOS data at pressures and temperatures where experimental data is difficult or impossible to obtain. The lack of experimental data, however, also precludes validation of the calculations in those regimes. Factors influencing the accuracy of first-principles data include theoretical approximations, and computational approximations used in implementing and solving the underlying equations. The first category includes approximate exchange-correlation functionals and wave equations simplifying the Dirac equation. In the second category are, e.g., basis completeness and pseudo-potentials. While the first category is extremely hard to assess without experimental data, inaccuracies of the second type should be well controlled. We are using two rather different electronic structure methods (VASP and RSPt) to make explicit the requirements for accuracy of the second type. We will discuss the VASP Projector Augmented Wave potentials, with examples for Li and Mo. Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000.

Numerical modeling of the interaction of liquid drops and jets with shock waves and gas jets

NASA Astrophysics Data System (ADS)

Surov, V. S.

1993-02-01

The motion of a liquid drop (jet) and of the ambient gas is described, in the general case, by Navier-Stokes equations. An approximate solution to the interaction of a plane shock wave with a single liquid drop is presented. Based on the analysis, the general system of Navier-Stokes equations is reduced to two groups of equations, Euler equations for gas and Navier-Stokes equations for liquid; solutions to these equations are presented. The discussion also covers the modeling of the interaction of a shock wave with a drop screen, interaction of a liquid jet with a counterpropagating supersonic gas flow, and modeling of processes in a shock layer during the impact of a drop against an obstacle in gas flow.

Partial differential equation-based localization of a monopole source from a circular array.

PubMed

Ando, Shigeru; Nara, Takaaki; Levy, Tsukassa

2013-10-01

Wave source localization from a sensor array has long been the most active research topics in both theory and application. In this paper, an explicit and time-domain inversion method for the direction and distance of a monopole source from a circular array is proposed. The approach is based on a mathematical technique, the weighted integral method, for signal/source parameter estimation. It begins with an exact form of the source-constraint partial differential equation that describes the unilateral propagation of wide-band waves from a single source, and leads to exact algebraic equations that include circular Fourier coefficients (phase mode measurements) as their coefficients. From them, nearly closed-form, single-shot and multishot algorithms are obtained that is suitable for use with band-pass/differential filter banks. Numerical evaluation and several experimental results obtained using a 16-element circular microphone array are presented to verify the validity of the proposed method.

The solution of non-linear hyperbolic equation systems by the finite element method

NASA Technical Reports Server (NTRS)

Loehner, R.; Morgan, K.; Zienkiewicz, O. C.

1984-01-01

A finite-element method for the solution of nonlinear hyperbolic systems of equations, such as those encountered in non-self-adjoint problems of transient phenomena in convection-diffusion or in the mixed representation of wave problems, is developed and demonstrated. The problem is rewritten in moving coordinates and reinterpolated to the original mesh by a Taylor expansion prior to a standard Galerkin spatial discretization, and it is shown that this procedure is equivalent to the time-discretization approach of Donea (1984). Numerical results for sample problems are presented graphically, including such shallow-water problems as the breaking of a dam, the shoaling of a wave, and the outflow of a river; compressible flows such as the isothermal flow in a nozzle and the Riemann shock-tube problem; and the two-dimensional scalar-advection, nonlinear-shallow-water, and Euler equations.

Elastic parabolic equation solutions for oceanic T-wave generation and propagation from deep seismic sources.

PubMed

Frank, Scott D; Collis, Jon M; Odom, Robert I

2015-06-01

Oceanic T-waves are earthquake signals that originate when elastic waves interact with the fluid-elastic interface at the ocean bottom and are converted to acoustic waves in the ocean. These waves propagate long distances in the Sound Fixing and Ranging (SOFAR) channel and tend to be the largest observed arrivals from seismic events. Thus, an understanding of their generation is important for event detection, localization, and source-type discrimination. Recently benchmarked seismic self-starting fields are used to generate elastic parabolic equation solutions that demonstrate generation and propagation of oceanic T-waves in range-dependent underwater acoustic environments. Both downward sloping and abyssal ocean range-dependent environments are considered, and results demonstrate conversion of elastic waves into water-borne oceanic T-waves. Examples demonstrating long-range broadband T-wave propagation in range-dependent environments are shown. These results confirm that elastic parabolic equation solutions are valuable for characterization of the relationships between T-wave propagation and variations in range-dependent bathymetry or elastic material parameters, as well as for modeling T-wave receptions at hydrophone arrays or coastal receiving stations.

Rogue wave spectra of the Kundu-Eckhaus equation.

PubMed

Bayındır, Cihan

2016-06-01

In this paper we analyze the rogue wave spectra of the Kundu-Eckhaus equation (KEE). We compare our findings with their nonlinear Schrödinger equation (NLSE) analogs and show that the spectra of the individual rogue waves significantly differ from their NLSE analogs. A remarkable difference is the one-sided development of the triangular spectrum before the rogue wave becomes evident in time. Also we show that increasing the skewness of the rogue wave results in increased asymmetry in the triangular Fourier spectra. Additionally, the triangular spectra of the rogue waves of the KEE begin to develop at earlier stages of their development compared to their NLSE analogs, especially for larger skew angles. This feature may be used to enhance the early warning times of the rogue waves. However, we show that in a chaotic wave field with many spectral components the triangular spectra remain as the main attribute as a universal feature of the typical wave fields produced through modulation instability and characteristic features of the KEE's analytical rogue wave spectra may be suppressed in a realistic chaotic wave field.

A novel unsplit perfectly matched layer for the second-order acoustic wave equation.

PubMed

Ma, Youneng; Yu, Jinhua; Wang, Yuanyuan

2014-08-01

When solving acoustic field equations by using numerical approximation technique, absorbing boundary conditions (ABCs) are widely used to truncate the simulation to a finite space. The perfectly matched layer (PML) technique has exhibited excellent absorbing efficiency as an ABC for the acoustic wave equation formulated as a first-order system. However, as the PML was originally designed for the first-order equation system, it cannot be applied to the second-order equation system directly. In this article, we aim to extend the unsplit PML to the second-order equation system. We developed an efficient unsplit implementation of PML for the second-order acoustic wave equation based on an auxiliary-differential-equation (ADE) scheme. The proposed method can benefit to the use of PML in simulations based on second-order equations. Compared with the existing PMLs, it has simpler implementation and requires less extra storage. Numerical results from finite-difference time-domain models are provided to illustrate the validity of the approach. Copyright © 2014 Elsevier B.V. All rights reserved.

On the dispersionless Kadomtsev-Petviashvili equation with arbitrary nonlinearity and dimensionality: exact solutions, longtime asymptotics of the Cauchy problem, wave breaking and shocks

NASA Astrophysics Data System (ADS)

Santucci, F.; Santini, P. M.

2016-10-01

We study the generalization of the dispersionless Kadomtsev-Petviashvili (dKP) equation in n+1 dimensions and with nonlinearity of degree m+1, a model equation describing the propagation of weakly nonlinear, quasi one-dimensional waves in the absence of dispersion and dissipation, and arising in several physical contexts, like acoustics, plasma physics, hydrodynamics and nonlinear optics. In 2 + 1 dimensions and with quadratic nonlinearity, this equation is integrable through a novel inverse scattering transform, and it has been recently shown to be a prototype model equation in the description of the two-dimensional wave breaking of localized initial data. In higher dimensions and with higher nonlinearity, the generalized dKP equations are not integrable, but their invariance under motions on the paraboloid allows one to construct in this paper a family of exact solutions describing waves constant on their paraboloidal wave front and breaking simultaneously in all points of it, developing after breaking either multivaluedness or single-valued discontinuous profiles (shocks). Then such exact solutions are used to build the longtime behavior of the solutions of the Cauchy problem, for small and localized initial data, showing that wave breaking of small initial data takes place in the longtime regime if and only if m(n-1)≤slant 2. Lastly, the analytic aspects of such wave breaking are investigated in detail in terms of the small initial data, in both cases in which the solution becomes multivalued after breaking or it develops a shock. These results, contained in the 2012 master’s thesis of one of the authors (FS) [1], generalize those obtained in [2] for the dKP equation in n+1 dimensions with quadratic nonlinearity, and are obtained following the same strategy.

On exact traveling-wave solutions for local fractional Korteweg-de Vries equation.

PubMed

Yang, Xiao-Jun; Tenreiro Machado, J A; Baleanu, Dumitru; Cattani, Carlo

2016-08-01

This paper investigates the Korteweg-de Vries equation within the scope of the local fractional derivative formulation. The exact traveling wave solutions of non-differentiable type with the generalized functions defined on Cantor sets are analyzed. The results for the non-differentiable solutions when fractal dimension is 1 are also discussed. It is shown that the exact solutions for the local fractional Korteweg-de Vries equation characterize the fractal wave on shallow water surfaces.

Perturbed soliton excitations of Rao-dust Alfvén waves in magnetized dusty plasmas

DOE Office of Scientific and Technical Information (OSTI.GOV)

Kavitha, L., E-mail: louiskavitha@yahoo.co.in; The Abdus Salam International Centre for Theoretical Physics, Trieste; Lavanya, C.

We investigate the propagation dynamics of the perturbed soliton excitations in a three component fully ionized dusty magnetoplasma consisting of electrons, ions, and heavy charged dust particulates. We derive the governing equation of motion for the two dimensional Rao-dust magnetohydrodynamic (R-D-MHD) wave by employing the inertialess electron equation of motion, inertial ion equation of motion, the continuity equations in a plasma with immobile charged dust grains, together with the Maxwell's equations, by assuming quasi neutrality and neglecting the displacement current in Ampere's law. Furthermore, we assume the massive dust particles are practically immobile since we are interested in timescales muchmore » shorter than the dusty plasma period, thereby neglecting any damping of the modes due to the grain charge fluctuations. We invoke the reductive perturbation method to represent the governing dynamics by a perturbed cubic nonlinear Schrödinger (pCNLS) equation. We solve the pCNLS, along the lines of Kodama-Ablowitz multiple scale nonlinear perturbation technique and explored the R-D-MHD waves as solitary wave excitations in a magnetized dusty plasma. Since Alfvén waves play an important role in energy transport in driving field-aligned currents, particle acceleration and heating, solar flares, and the solar wind, this representation of R-D-MHD waves as soliton excitations may have extensive applications to study the lower part of the earth's ionosphere.« less

The Gassmann-Burgers Model to Simulate Seismic Waves at the Earth Crust And Mantle

NASA Astrophysics Data System (ADS)

Carcione, José M.; Poletto, Flavio; Farina, Biancamaria; Craglietto, Aronne

2017-03-01

The upper part of the crust shows generally brittle behaviour while deeper zones, including the mantle, may present ductile behaviour, depending on the pressure-temperature conditions; moreover, some parts are melted. Seismic waves can be used to detect these conditions on the basis of reflection and transmission events. Basically, from the elastic-plastic point of view the seismic properties (seismic velocity and density) depend on effective pressure and temperature. Confining and pore pressures have opposite effects on these properties, such that very small effective pressures (the presence of overpressured fluids) may substantially decrease the P- and S-wave velocities, mainly the latter, by opening of cracks and weakening of grain contacts. Similarly, high temperatures induce the same effect by partial melting. To model these effects, we consider a poro-viscoelastic model based on Gassmann equations and Burgers mechanical model to represent the properties of the rock frame and describe ductility in which deformation takes place by shear plastic flow. The Burgers elements allow us to model the effects of seismic attenuation, velocity dispersion and steady-state creep flow, respectively. The stiffness components of the brittle and ductile media depend on stress and temperature through the shear viscosity, which is obtained by the Arrhenius equation and the octahedral stress criterion. Effective pressure effects are taken into account in the dry-rock moduli using exponential functions whose parameters are obtained by fitting experimental data as a function of confining pressure. Since fluid effects are important, the density and bulk modulus of the saturating fluids (water and steam) are modeled using the equations provided by the NIST website, including supercritical behaviour. The theory allows us to obtain the phase velocity and quality factor as a function of depth and geological pressure and temperature as well as time frequency. We then obtain the PS and SH equations of motion recast in the velocity-stress formulation, including memory variables to avoid the computation of time convolutions. The equations correspond to isotropic anelastic and inhomogeneous media and are solved by a direct grid method based on the Runge-Kutta time stepping technique and the Fourier pseudospectral method. The algorithm is tested with success against known analytical solutions for different shear viscosities. An example shows how anomalous conditions of pressure and temperature can in principle be detected with seismic waves.

Prestack reverse time migration for tilted transversely isotropic media

NASA Astrophysics Data System (ADS)

Jang, Seonghyung; Hien, Doan Huy

2013-04-01

According to having interest in unconventional resource plays, anisotropy problem is naturally considered as an important step for improving the seismic image quality. Although it is well known prestack depth migration for the seismic reflection data is currently one of the powerful tools for imaging complex geological structures, it may lead to migration error without considering anisotropy. Asymptotic analysis of wave propagation in transversely isotropic (TI) media yields a dispersion relation of couple P- and SV wave modes that can be converted to a fourth order scalar partial differential equation (PDE). By setting the shear wave velocity equal zero, the fourth order PDE, called an acoustic wave equation for TI media, can be reduced to couple of second order PDE systems and we try to solve the second order PDE by the finite difference method (FDM). The result of this P wavefield simulation is kinematically similar to elastic and anisotropic wavefield simulation. We develop prestack depth migration algorithm for tilted transversely isotropic media using reverse time migration method (RTM). RTM is a method for imaging the subsurface using inner product of source wavefield extrapolation in forward and receiver wavefield extrapolation in backward. We show the subsurface image in TTI media using the inner product of partial derivative wavefield with respect to physical parameters and observation data. Since the partial derivative wavefields with respect to the physical parameters require extremely huge computing time, so we implemented the imaging condition by zero lag crosscorrelation of virtual source and back propagating wavefield instead of partial derivative wavefields. The virtual source is calculated directly by solving anisotropic acoustic wave equation, the back propagating wavefield on the other hand is calculated by the shot gather used as the source function in the anisotropic acoustic wave equation. According to the numerical model test for a simple geological model including syncline and anticline, the prestack depth migration using TTI-RTM in weak anisotropic media shows the subsurface image is similar to the true geological model used to generate the shot gathers.

Regular and singular pulse and front solutions and possible isochronous behavior in the short-pulse equation: Phase-plane, multi-infinite series and variational approaches

NASA Astrophysics Data System (ADS)

Gambino, G.; Tanriver, U.; Guha, P.; Choudhury, A. Ghose; Choudhury, S. Roy

2015-02-01

In this paper we employ three recent analytical approaches to investigate the possible classes of traveling wave solutions of some members of a family of so-called short-pulse equations (SPE). A recent, novel application of phase-plane analysis is first employed to show the existence of breaking kink wave solutions in certain parameter regimes. Secondly, smooth traveling waves are derived using a recent technique to derive convergent multi-infinite series solutions for the homoclinic (heteroclinic) orbits of the traveling-wave equations for the SPE equation, as well as for its generalized version with arbitrary coefficients. These correspond to pulse (kink or shock) solutions respectively of the original PDEs. We perform many numerical tests in different parameter regime to pinpoint real saddle equilibrium points of the corresponding traveling-wave equations, as well as ensure simultaneous convergence and continuity of the multi-infinite series solutions for the homoclinic/heteroclinic orbits anchored by these saddle points. Unlike the majority of unaccelerated convergent series, high accuracy is attained with relatively few terms. And finally, variational methods are employed to generate families of both regular and embedded solitary wave solutions for the SPE PDE. The technique for obtaining the embedded solitons incorporates several recent generalizations of the usual variational technique and it is thus topical in itself. One unusual feature of the solitary waves derived here is that we are able to obtain them in analytical form (within the assumed ansatz for the trial functions). Thus, a direct error analysis is performed, showing the accuracy of the resulting solitary waves. Given the importance of solitary wave solutions in wave dynamics and information propagation in nonlinear PDEs, as well as the fact that not much is known about solutions of the family of generalized SPE equations considered here, the results obtained are both new and timely.

Low-frequency fluid waves in fractures and pipes

DOE Office of Scientific and Technical Information (OSTI.GOV)

Korneev, Valeri

2010-09-01

Low-frequency analytical solutions have been obtained for phase velocities of symmetrical fluid waves within both an infinite fracture and a pipe filled with a viscous fluid. Three different fluid wave regimes can exist in such objects, depending on the various combinations of parameters, such as fluid density, fluid viscosity, walls shear modulus, channel thickness, and frequency. Equations for velocities of all these regimes have explicit forms and are verified by comparisons with the exact solutions. The dominant role of fractures in rock permeability at field scales and the strong amplitude and frequency effects of Stoneley guided waves suggest the importancemore » of including these wave effects into poroelastic theories.« less

Feasibility of detecting near-surface feature with Rayleigh-wave diffraction

USGS Publications Warehouse

Xia, J.; Nyquist, Jonathan E.; Xu, Y.; Roth, M.J.S.; Miller, R.D.

2007-01-01

Detection of near-surfaces features such as voids and faults is challenging due to the complexity of near-surface materials and the limited resolution of geophysical methods. Although multichannel, high-frequency, surface-wave techniques can provide reliable shear (S)-wave velocities in different geological settings, they are not suitable for detecting voids directly based on anomalies of the S-wave velocity because of limitations on the resolution of S-wave velocity profiles inverted from surface-wave phase velocities. Therefore, we studied the feasibility of directly detecting near-surfaces features with surface-wave diffractions. Based on the properties of surface waves, we have derived a Rayleigh-wave diffraction traveltime equation. We also have solved the equation for the depth to the top of a void and an average velocity of Rayleigh waves. Using these equations, the depth to the top of a void/fault can be determined based on traveltime data from a diffraction curve. In practice, only two diffraction times are necessary to define the depth to the top of a void/fault and the average Rayleigh-wave velocity that generates the diffraction curve. We used four two-dimensional square voids to demonstrate the feasibility of detecting a void with Rayleigh-wave diffractions: a 2??m by 2??m with a depth to the top of the void of 2??m, 4??m by 4??m with a depth to the top of the void of 7??m, and 6??m by 6??m with depths to the top of the void 12??m and 17??m. We also modeled surface waves due to a vertical fault. Rayleigh-wave diffractions were recognizable for all these models after FK filtering was applied to the synthetic data. The Rayleigh-wave diffraction traveltime equation was verified by the modeled data. Modeling results suggested that FK filtering is critical to enhance diffracted surface waves. A real-world example is presented to show how to utilize the derived equation of surface-wave diffractions. ?? 2006 Elsevier B.V. All rights reserved.

Solitary wave for a nonintegrable discrete nonlinear Schrödinger equation in nonlinear optical waveguide arrays

NASA Astrophysics Data System (ADS)

Ma, Li-Yuan; Ji, Jia-Liang; Xu, Zong-Wei; Zhu, Zuo-Nong

2018-03-01

We study a nonintegrable discrete nonlinear Schrödinger (dNLS) equation with the term of nonlinear nearest-neighbor interaction occurred in nonlinear optical waveguide arrays. By using discrete Fourier transformation, we obtain numerical approximations of stationary and travelling solitary wave solutions of the nonintegrable dNLS equation. The analysis of stability of stationary solitary waves is performed. It is shown that the nonlinear nearest-neighbor interaction term has great influence on the form of solitary wave. The shape of solitary wave is important in the electric field propagating. If we neglect the nonlinear nearest-neighbor interaction term, much important information in the electric field propagating may be missed. Our numerical simulation also demonstrates the difference of chaos phenomenon between the nonintegrable dNLS equation with nonlinear nearest-neighbor interaction and another nonintegrable dNLS equation without the term. Project supported by the National Natural Science Foundation of China (Grant Nos. 11671255 and 11701510), the Ministry of Economy and Competitiveness of Spain (Grant No. MTM2016-80276-P (AEI/FEDER, EU)), and the China Postdoctoral Science Foundation (Grant No. 2017M621964).

Soliton interactions and Bäcklund transformation for a (2+1)-dimensional variable-coefficient modified Kadomtsev-Petviashvili equation in fluid dynamics

NASA Astrophysics Data System (ADS)

Xiao, Zi-Jian; Tian, Bo; Sun, Yan

2018-01-01

In this paper, we investigate a (2+1)-dimensional variable-coefficient modified Kadomtsev-Petviashvili (mKP) equation in fluid dynamics. With the binary Bell-polynomial and an auxiliary function, bilinear forms for the equation are constructed. Based on the bilinear forms, multi-soliton solutions and Bell-polynomial-type Bäcklund transformation for such an equation are obtained through the symbolic computation. Soliton interactions are presented. Based on the graphic analysis, Parametric conditions for the existence of the shock waves, elevation solitons and depression solitons are given, and it is shown that under the condition of keeping the wave vectors invariable, the change of α(t) and β(t) can lead to the change of the solitonic velocities, but the shape of each soliton remains unchanged, where α(t) and β(t) are the variable coefficients in the equation. Oblique elastic interactions can exist between the (i) two shock waves, (ii) two elevation solitons, and (iii) elevation and depression solitons. However, oblique interactions between (i) shock waves and elevation solitons, (ii) shock waves and depression solitons are inelastic.

Application of the Parabolic Approximation to Predict Acoustical Propagation in the Ocean.

ERIC Educational Resources Information Center

McDaniel, Suzanne T.

1979-01-01

A simplified derivation of the parabolic approximation to the acoustical wave equation is presented. Exact solutions to this approximate equation are compared with solutions to the wave equation to demonstrate the applicability of this method to the study of underwater sound propagation. (Author/BB)

Capillary waves in the subcritical nonlinear Schroedinger equation

DOE Office of Scientific and Technical Information (OSTI.GOV)

Kozyreff, G.

2010-01-15

We expand recent results on the nonlinear Schroedinger equation with cubic-quintic nonlinearity to show that some solutions are described by the Bernoulli equation in the presence of surface tension. As a consequence, capillary waves are predicted and found numerically at the interface between regions of large and low amplitude.

WavePacket: A Matlab package for numerical quantum dynamics. I: Closed quantum systems and discrete variable representations

NASA Astrophysics Data System (ADS)

Schmidt, Burkhard; Lorenz, Ulf

2017-04-01

WavePacket is an open-source program package for the numerical simulation of quantum-mechanical dynamics. It can be used to solve time-independent or time-dependent linear Schrödinger and Liouville-von Neumann-equations in one or more dimensions. Also coupled equations can be treated, which allows to simulate molecular quantum dynamics beyond the Born-Oppenheimer approximation. Optionally accounting for the interaction with external electric fields within the semiclassical dipole approximation, WavePacket can be used to simulate experiments involving tailored light pulses in photo-induced physics or chemistry. The graphical capabilities allow visualization of quantum dynamics 'on the fly', including Wigner phase space representations. Being easy to use and highly versatile, WavePacket is well suited for the teaching of quantum mechanics as well as for research projects in atomic, molecular and optical physics or in physical or theoretical chemistry. The present Part I deals with the description of closed quantum systems in terms of Schrödinger equations. The emphasis is on discrete variable representations for spatial discretization as well as various techniques for temporal discretization. The upcoming Part II will focus on open quantum systems and dimension reduction; it also describes the codes for optimal control of quantum dynamics. The present work introduces the MATLAB version of WavePacket 5.2.1 which is hosted at the Sourceforge platform, where extensive Wiki-documentation as well as worked-out demonstration examples can be found.

High-order upwind schemes for the wave equation on overlapping grids: Maxwell's equations in second-order form

DOE Office of Scientific and Technical Information (OSTI.GOV)

Angel, Jordan B.; Banks, Jeffrey W.; Henshaw, William D.

High-order accurate upwind approximations for the wave equation in second-order form on overlapping grids are developed. Although upwind schemes are well established for first-order hyperbolic systems, it was only recently shown by Banks and Henshaw how upwinding could be incorporated into the second-order form of the wave equation. This new upwind approach is extended here to solve the time-domain Maxwell's equations in second-order form; schemes of arbitrary order of accuracy are formulated for general curvilinear grids. Taylor time-stepping is used to develop single-step space-time schemes, and the upwind dissipation is incorporated by embedding the exact solution of a local Riemannmore » problem into the discretization. Second-order and fourth-order accurate schemes are implemented for problems in two and three space dimensions, and overlapping grids are used to treat complex geometry and problems with multiple materials. Stability analysis of the upwind-scheme on overlapping grids is performed using normal mode theory. The stability analysis and computations confirm that the upwind scheme remains stable on overlapping grids, including the difficult case of thin boundary grids when the traditional non-dissipative scheme becomes unstable. The accuracy properties of the scheme are carefully evaluated on a series of classical scattering problems for both perfect conductors and dielectric materials in two and three space dimensions. Finally, the upwind scheme is shown to be robust and provide high-order accuracy.« less

High-order upwind schemes for the wave equation on overlapping grids: Maxwell's equations in second-order form

DOE PAGES

Angel, Jordan B.; Banks, Jeffrey W.; Henshaw, William D.

2017-09-28

High-order accurate upwind approximations for the wave equation in second-order form on overlapping grids are developed. Although upwind schemes are well established for first-order hyperbolic systems, it was only recently shown by Banks and Henshaw how upwinding could be incorporated into the second-order form of the wave equation. This new upwind approach is extended here to solve the time-domain Maxwell's equations in second-order form; schemes of arbitrary order of accuracy are formulated for general curvilinear grids. Taylor time-stepping is used to develop single-step space-time schemes, and the upwind dissipation is incorporated by embedding the exact solution of a local Riemannmore » problem into the discretization. Second-order and fourth-order accurate schemes are implemented for problems in two and three space dimensions, and overlapping grids are used to treat complex geometry and problems with multiple materials. Stability analysis of the upwind-scheme on overlapping grids is performed using normal mode theory. The stability analysis and computations confirm that the upwind scheme remains stable on overlapping grids, including the difficult case of thin boundary grids when the traditional non-dissipative scheme becomes unstable. The accuracy properties of the scheme are carefully evaluated on a series of classical scattering problems for both perfect conductors and dielectric materials in two and three space dimensions. Finally, the upwind scheme is shown to be robust and provide high-order accuracy.« less

High-order upwind schemes for the wave equation on overlapping grids: Maxwell's equations in second-order form

NASA Astrophysics Data System (ADS)

Angel, Jordan B.; Banks, Jeffrey W.; Henshaw, William D.

2018-01-01

High-order accurate upwind approximations for the wave equation in second-order form on overlapping grids are developed. Although upwind schemes are well established for first-order hyperbolic systems, it was only recently shown by Banks and Henshaw [1] how upwinding could be incorporated into the second-order form of the wave equation. This new upwind approach is extended here to solve the time-domain Maxwell's equations in second-order form; schemes of arbitrary order of accuracy are formulated for general curvilinear grids. Taylor time-stepping is used to develop single-step space-time schemes, and the upwind dissipation is incorporated by embedding the exact solution of a local Riemann problem into the discretization. Second-order and fourth-order accurate schemes are implemented for problems in two and three space dimensions, and overlapping grids are used to treat complex geometry and problems with multiple materials. Stability analysis of the upwind-scheme on overlapping grids is performed using normal mode theory. The stability analysis and computations confirm that the upwind scheme remains stable on overlapping grids, including the difficult case of thin boundary grids when the traditional non-dissipative scheme becomes unstable. The accuracy properties of the scheme are carefully evaluated on a series of classical scattering problems for both perfect conductors and dielectric materials in two and three space dimensions. The upwind scheme is shown to be robust and provide high-order accuracy.

Approximating a nonlinear advanced-delayed equation from acoustics

NASA Astrophysics Data System (ADS)

Teodoro, M. Filomena

2016-10-01

We approximate the solution of a particular non-linear mixed type functional differential equation from physiology, the mucosal wave model of the vocal oscillation during phonation. The mathematical equation models a superficial wave propagating through the tissues. The numerical scheme is adapted from the work presented in [1, 2, 3], using homotopy analysis method (HAM) to solve the non linear mixed type equation under study.

Electromagnetic pulses, localized and causal

NASA Astrophysics Data System (ADS)

Lekner, John

2018-01-01

We show that pulse solutions of the wave equation can be expressed as time Fourier superpositions of scalar monochromatic beam wave functions (solutions of the Helmholtz equation). This formulation is shown to be equivalent to Bateman's integral expression for solutions of the wave equation, for axially symmetric solutions. A closed-form one-parameter solution of the wave equation, containing no backward-propagating parts, is constructed from a beam which is the tight-focus limit of two families of beams. Application is made to transverse electric and transverse magnetic pulses, with evaluation of the energy, momentum and angular momentum for a pulse based on the general localized and causal form. Such pulses can be represented as superpositions of photons. Explicit total energy and total momentum values are given for the one-parameter closed-form pulse.

Fifth-order complex Korteweg-de Vries-type equations

NASA Astrophysics Data System (ADS)

Khanal, Netra; Wu, Jiahong; Yuan, Juan-Ming

2012-05-01

This paper studies spatially periodic complex-valued solutions of the fifth-order Korteweg-de Vries (KdV)-type equations. The aim is at several fundamental issues including the existence, uniqueness and finite-time blowup problems. Special attention is paid to the Kawahara equation, a fifth-order KdV-type equation. When a Burgers dissipation is attached to the Kawahara equation, we establish the existence and uniqueness of the Fourier series solution with the Fourier modes decaying algebraically in terms of the wave numbers. We also examine a special series solution to the Kawahara equation and prove the convergence and global regularity of such solutions associated with a single mode initial data. In addition, finite-time blowup results are discussed for the special series solution of the Kawahara equation.

Unsteady free surface flow in porous media: One-dimensional model equations including vertical effects and seepage face

NASA Astrophysics Data System (ADS)

Di Nucci, Carmine

2018-05-01

This note examines the two-dimensional unsteady isothermal free surface flow of an incompressible fluid in a non-deformable, homogeneous, isotropic, and saturated porous medium (with zero recharge and neglecting capillary effects). Coupling a Boussinesq-type model for nonlinear water waves with Darcy's law, the two-dimensional flow problem is solved using one-dimensional model equations including vertical effects and seepage face. In order to take into account the seepage face development, the system equations (given by the continuity and momentum equations) are completed by an integral relation (deduced from the Cauchy theorem). After testing the model against data sets available in the literature, some numerical simulations, concerning the unsteady flow through a rectangular dam (with an impermeable horizontal bottom), are presented and discussed.

Full wave description of VLF wave penetration through the ionosphere

NASA Astrophysics Data System (ADS)

Kuzichev, Ilya; Shklyar, David

2010-05-01

Of the many problems in whistler study, wave propagation through the ionosphere is among the most important, and the most difficult at the same time. Both satellite and ground-based investigations of VLF waves include considerations of this problem, and it has been in the focus of research since the beginning of whistler study (Budden [1985]; Helliwell [1965]). The difficulty in considering VLF wave passage through the ionosphere is, after all, due to fast variation of the lower ionosphere parameters as compared to typical VLF wave number. This makes irrelevant the consideration in the framework of geometrical optics, which, along with a smooth variations of parameters, is always based on a particular dispersion relation. Although the full wave analysis in the framework of cold plasma approximation does not require slow variations of plasma parameters, and does not assume any particular wave mode, the fact that the wave of a given frequency belongs to different modes in various regions makes numerical solution of the field equations not simple. More specifically, as is well known (e.g. Ginzburg and Rukhadze [1972]), in a cold magnetized plasma, there are, in general, two wave modes related to a given frequency. Both modes, however, do not necessarily correspond to propagating waves. In particular, in the frequency range related to whistler waves, the other mode is evanescent, i.e. it has a negative value of N2 (the refractive index squared). It means that one of solutions of the relevant differential equations is exponentially growing, which makes a straightforward numerical approach to these equations despairing. This well known difficulty in the problem under discussion is usually identified as numerical swamping (Budden [1985]). Resolving the problem of numerical swamping becomes, in fact, a key point in numerical study of wave passage through the ionosphere. As it is typical of work based on numerical simulations, its essential part remains virtually hidden. Then, every researcher, in order to get quantitative characteristics of the process, such as transmission and reflection coefficients, needs to go through the whole problem. That is why the number of publications dealing with VLF wave transmission through the ionosphere does not run short. In this work, we develop a new approach to the problem, such that its intrinsic difficulty is resolved analytically, while numerical calculations are reduced to stable equations solvable with the help of a routine program. Using this approach, the field of VLF wave incident on the ionosphere from above is calculated as a function of height, and reflection coefficients for different frequencies and angles of incidence are obtained. In particular, for small angles of incidence, for which incident waves reach the ground, the reflection coefficient appears to be an oscillating function of frequency. Another goal of the work is to present all equations and related formulae in an undisguised form, in order that the problem may be solved in a straightforward way, once the ionospheric plasma parameters are given. References Budden, K.G. (1985), The Propagation of Radio Waves, Cambridge Univ. Press, Cambridge, U.K. Ginzburg, V.L., and Rukhadze, A.A. (1972), Waves in Magnetoactive Plasma. In Handbuch der Physik (ed. S. Flügge). Vol. 49, Part IV, p. 395, Springer Verlag, Berlin. Helliwell, R. A. (1965), Whistlers and Related Ionospheric Phenomena, Stanford University Press, Stanford, California.

Reflection coefficient of qP, qS and SH at a plane boundary between viscoelastic TTI media

NASA Astrophysics Data System (ADS)

Wang, Hongwei; Peng, Suping

2016-01-01

This paper introduces a calculation method for the effective elastic stiffness tensor matrix of the viscous-elastic TTI medium based on the Chapman theory. We then obtain the phase velocity formula and seismic wave polarization formula of the viscous-elastic TTI medium, by solving the Christoffel equation; solve the phase angle of reflection and transmission wave through the numerical method in accordance with the wave slowness ellipsoid; on the basis of this assumption, and assuming that qP, qS and SH waves occurred simultaneously at the viscous-elastic anisotropic interface, establish the sixth-order Zoeppritz equation in accordance with the boundary conditions; establish the models for the upper and lower media which are viscous-elastic HTI, TTI, etc., on the basis of the sixth-order Zoeppritz equation; and study the impact of fracture dip angle, azimuth angle and frequency on the reflection coefficient. From this we obtain the following conclusions: the reflection coefficient can identify the fracture strike and dip when any information pertaining to the media is unknown; dispersion phenomenon is obvious on the axial plane of symmetry and weakened in the plane vertical to the axial plane of symmetry; the vertical-incidence longitudinal wave can stimulate the qS wave when the dip angle is not 0° or 90° under the condition of coincidence between the symmetry planes of the upper and lower media; when the symmetry planes of the upper and lower media do not coincide and the dip angle is not 0° or 90°, then the vertical-incidence qP will stimulate the qS and SH waves at the same time; the dip angle can cause the reflection coefficient curve to have a more obvious dispersion phenomenon, while the included angle between the symmetry planes of the upper and lower media will weaken the dispersion except SH; and the intercept of reflection coefficient is affected by the fracture dip and included angle between the symmetry planes of the upper and lower media.

Hamiltonian structures for systems of hyperbolic conservation laws

NASA Astrophysics Data System (ADS)

Olver, Peter J.; Nutku, Yavuz

1988-07-01

The bi-Hamiltonian structure for a large class of one-dimensional hyberbolic systems of conservation laws in two field variables, including the equations of gas dynamics, shallow water waves, one-dimensional elastic media, and the Born-Infeld equation from nonlinear electrodynamics, is exhibited. For polytropic gas dynamics, these results lead to a quadri-Hamiltonian structure. New higher-order entropy-flux pairs (conservation laws) and higher-order symmetries are exhibited.

Numerical modelling of multiphase liquid-vapor-gas flows with interfaces and cavitation

NASA Astrophysics Data System (ADS)

Pelanti, Marica

2017-11-01

We are interested in the simulation of multiphase flows where the dynamical appearance of vapor cavities and evaporation fronts in a liquid is coupled to the dynamics of a third non-condensable gaseous phase. We describe these flows by a single-velocity three-phase compressible flow model composed of the phasic mass and total energy equations, the volume fraction equations, and the mixture momentum equation. The model includes stiff mechanical and thermal relaxation source terms for all the phases, and chemical relaxation terms to describe mass transfer between the liquid and vapor phases of the species that may undergo transition. The flow equations are solved by a mixture-energy-consistent finite volume wave propagation scheme, combined with simple and robust procedures for the treatment of the stiff relaxation terms. An analytical study of the characteristic wave speeds of the hierarchy of relaxed models associated to the parent model system is also presented. We show several numerical experiments, including two-dimensional simulations of underwater explosive phenomena where highly pressurized gases trigger cavitation processes close to a rigid surface or to a free surface. This work was supported by the French Government Grant DGA N. 2012.60.0011.00.470.75.01, and partially by the Norwegian Grant RCN N. 234126/E30.

An efficient hybrid pseudospectral/finite-difference scheme for solving the TTI pure P-wave equation

NASA Astrophysics Data System (ADS)

Zhan, Ge; Pestana, Reynam C.; Stoffa, Paul L.

2013-04-01

The pure P-wave equation for modelling and migration in tilted transversely isotropic (TTI) media has attracted more and more attention in imaging seismic data with anisotropy. The desirable feature is that it is absolutely free of shear-wave artefacts and the consequent alleviation of numerical instabilities generally suffered by some systems of coupled equations. However, due to several forward-backward Fourier transforms in wavefield updating at each time step, the computational cost is significant, and thereby hampers its prevalence. We propose to use a hybrid pseudospectral (PS) and finite-difference (FD) scheme to solve the pure P-wave equation. In the hybrid solution, most of the cost-consuming wavenumber terms in the equation are replaced by inexpensive FD operators, which in turn accelerates the computation and reduces the computational cost. To demonstrate the benefit in cost saving of the new scheme, 2D and 3D reverse-time migration (RTM) examples using the hybrid solution to the pure P-wave equation are carried out, and respective runtimes are listed and compared. Numerical results show that the hybrid strategy demands less computation time and is faster than using the PS method alone. Furthermore, this new TTI RTM algorithm with the hybrid method is computationally less expensive than that with the FD solution to conventional TTI coupled equations.

Nonlinear equations of motion for Landau resonance interactions with a whistler mode wave

NASA Technical Reports Server (NTRS)

Inan, U. S.; Tkalcevic, S.

1982-01-01

A simple set of equations is presented for the description of the cyclotron averaged motion of Landau resonant particles in a whistler mode wave propagating at an angle to the static magnetic field. A comparison is conducted of the wave magnetic field and electric field effects for the parameters of the magnetosphere, and the parameter ranges for which the wave magnetic field effects would be negligible are determined. It is shown that the effect of the wave magnetic field can be neglected for low pitch angles, high normal wave angles, and/or high normalized wave frequencies.

Gravitational waves, energy and Feynman’s “sticky bead”

NASA Astrophysics Data System (ADS)

Cooperstock, F. I.

2015-07-01

It is noted that in the broader sense, gravitational waves viewed as spacetime curvature which necessarily accompanies electromagnetic waves at the speed of light, are the routine perception of our everyday experience. We focus on the energy issue and Feynman’s “sticky bead” argument which has been regarded as central in supporting the conclusion that gravitational waves carry energy through the vacuum in general relativity. We discuss the essential neglected aspects of his approach which leads to the conclusion that gravitational waves would not cause Feynman’s bead to heat the stick on which it would supposedly rub. This opens the way to an examination of the entire issue of energy in general relativity. We briefly discuss our naturally-defined totally invariant spacetime energy expression for general relativity incorporating the contribution from gravity. When the cosmological term is included in the field equations, our energy expression includes the vacuum energy as required.

Analytical study of laser-supported combustion waves in hydrogen

NASA Technical Reports Server (NTRS)

Kemp, N. H.; Root, R. G.

1978-01-01

Laser supported combustion (LSC) waves are an important ingredient in the fluid mechanics of CW laser propulsion using a hydrogen propellant and 10.6 micron lasers. Therefore, a computer model has been constructed to solve the one-dimensional energy equation with constant pressure and area. Physical processes considered include convection, conduction, absorption of laser energy, radiation energy loss, and accurate properties of equilibrium hydrogen. Calculations for 1, 3, 10 and 30 atm were made for intensities of 10 to the 4th to 10 to the 6th W/sq cm, which gave temperature profiles, wave speed, etc. To pursue the propulsion application, a second computer model was developed to describe the acceleration of the gas emerging from the LSC wave into a variable-pressure, converging streamtube, still including all the above-mentioned physical processes. The results show very high temperatures in LSC waves which absorb all the laser energy, and high radiative losses.

Nonlinear modulation of an extraordinary wave under the conditions of parametric decay

DOE Office of Scientific and Technical Information (OSTI.GOV)

Dorofeenko, V. G.; Krasovitskiy, V. B.; Turikov, V. A.

2012-06-15

A self-consistent set of Hamilton equations describing nonlinear saturation of the amplitude of oscillations excited under the conditions of parametric decay of an elliptically polarized extraordinary wave in cold plasma is solved analytically and numerically. It is shown that the exponential increase in the amplitude of the secondary wave excited at the half-frequency of the primary wave changes into a reverse process in which energy is returned to the primary wave and nonlinear oscillations propagating across the external magnetic field are generated. The system of 'slow' equations for the amplitudes, obtained by averaging the initial equations over the high-frequency period,more » is used to describe steady-state nonlinear oscillations in plasma.« less

DOE Office of Scientific and Technical Information (OSTI.GOV)

Sano, Yukio; Abe, Akihisa; Tokushima, Koji

The aim of this study is to examine the difference between shock temperatures predicted by an equation for temperature inside a steady wave front and the Walsh-Christian equation. Calculations are for yttria-doped tetragonal zirconia, which shows an elastic-plastic and a phase transition: Thus the shock waves treated are multiple structure waves composed of one to three steady wave fronts. The evaluated temperature was 3350K at the minimum specific volume of 0.1175 cm{sup 3}/g (or maximum Hugoniot shock pressure of 140GPa) considered in the present examination, while the temperature predicted by the Walsh-Christian equation under identical conditions was 2657K. The causemore » of the large temperature discrepancy is considered to be that the present model treats nonequilibrium states inside steady waves.« less

Rogue waves in multiple-solitons-inelastic collisions — The complex Sharma-Tasso-Olver equation

NASA Astrophysics Data System (ADS)

Abdel-Gawad, H. I.; Tantawy, M.

2018-03-01

Very recently, a mechanism to the formation of rogue waves (RWs) has been proposed by the authors. In this paper, the formation of RWs in case of the complex Sharma-Tasso-Olver (STO) equation is studied. In the STO equation, one, two and three-soliton solutions are obtained. Due to the inelastic collisions, these soliton waves are fused to one. Under the free parameters constraint this behavior do occurs. The mechanism of formation of RWs is due to the collisions of solitons and multi-periodic waves (like spectral band). These RWs as giant waves, which may be very sharp or chaotic are similar to RWs in laser. The work is done here by using the generalized unified method (GUM).

Calculation of the Full Scattering Amplitude without Partial Wave Decomposition. 2; Inclusion of Exchange

NASA Technical Reports Server (NTRS)

Shertzer, Janine; Temkin, Aaron

2004-01-01

The development of a practical method of accurately calculating the full scattering amplitude, without making a partial wave decomposition is continued. The method is developed in the context of electron-hydrogen scattering, and here exchange is dealt with by considering e-H scattering in the static exchange approximation. The Schroedinger equation in this approximation can be simplified to a set of coupled integro-differential equations. The equations are solved numerically for the full scattering wave function. The scattering amplitude can most accurately be calculated from an integral expression for the amplitude; that integral can be formally simplified, and then evaluated using the numerically determined wave function. The results are essentially identical to converged partial wave results.

Properties of Nonlinear Dynamo Waves

NASA Technical Reports Server (NTRS)

Tobias, S. M.

1997-01-01

Dynamo theory offers the most promising explanation of the generation of the sun's magnetic cycle. Mean field electrodynamics has provided the platform for linear and nonlinear models of solar dynamos. However, the nonlinearities included are (necessarily) arbitrarily imposed in these models. This paper conducts a systematic survey of the role of nonlinearities in the dynamo process, by considering the behaviour of dynamo waves in the nonlinear regime. It is demonstrated that only by considering realistic nonlinearities that are non-local in space and time can modulation of the basic dynamo wave he achieved. Moreover, this modulation is greatest when there is a large separation of timescales provided by including a low magnetic Prandtl number in the equation for the velocity perturbations.

Soliton's eigenvalue based analysis on the generation mechanism of rogue wave phenomenon in optical fibers exhibiting weak third order dispersion.

PubMed

Weerasekara, Gihan; Tokunaga, Akihiro; Terauchi, Hiroki; Eberhard, Marc; Maruta, Akihiro

2015-01-12

One of the extraordinary aspects of nonlinear wave evolution which has been observed as the spontaneous occurrence of astonishing and statistically extraordinary amplitude wave is called rogue wave. We show that the eigenvalues of the associated equation of nonlinear Schrödinger equation are almost constant in the vicinity of rogue wave and we validate that optical rogue waves are formed by the collision between quasi-solitons in anomalous dispersion fiber exhibiting weak third order dispersion.

Upstream-advancing waves generated by three-dimensional moving disturbances

NASA Astrophysics Data System (ADS)

Lee, Seung-Joon; Grimshaw, Roger H. J.

1990-02-01

The wave field resulting from a surface pressure or a bottom topography in a horizontally unbounded domain is studied. Upstream-advancing waves successively generated by various forcing disturbances moving with near-resonant speeds are found by numerically solving a forced Kadomtsev-Petviashvili (fKP) equation, which shows in its simplest form the interplay of a basic linear wave operator, longitudinal and transverse dispersion, nonlinearity, and forcing. Curved solitary waves are found as a slowly varying similarity solution of the Kadomtsev-Petviashvili (KP) equation, and are favorably compared with the upstream-advancing waves numerically obtained.

Freak waves in random oceanic sea states.

PubMed

Onorato, M; Osborne, A R; Serio, M; Bertone, S

2001-06-18

Freak waves are very large, rare events in a random ocean wave train. Here we study their generation in a random sea state characterized by the Joint North Sea Wave Project spectrum. We assume, to cubic order in nonlinearity, that the wave dynamics are governed by the nonlinear Schrödinger (NLS) equation. We show from extensive numerical simulations of the NLS equation how freak waves in a random sea state are more likely to occur for large values of the Phillips parameter alpha and the enhancement coefficient gamma. Comparison with linear simulations is also reported.

An Analytical Model of Periodic Waves in Shallow Water--Summary.

DTIC Science & Technology

1984-01-01

Petviashvili equation , and is based on a Riemann theta function of genus 2. These bi-periodic waves are direct generalizations of the well-known (simply... Petviashvili (KP; 1970) equation , (ut 6uux + U ) 3uyy -0, (1) is a scaled, dimensionless equation that describes the evolution of long water waves of...Fluid Mech., vol. 92, pp 691-715 Dubrovin, B. A., 1981, Russ. Math. Surveys, vol. 36, pp 11-92 Kadomtsev , B. B. & V. I. Petviashvili , 1970,) Soy. Phys

Infinite hierarchy of nonlinear Schrödinger equations and their solutions.

PubMed

Ankiewicz, A; Kedziora, D J; Chowdury, A; Bandelow, U; Akhmediev, N

2016-01-01

We study the infinite integrable nonlinear Schrödinger equation hierarchy beyond the Lakshmanan-Porsezian-Daniel equation which is a particular (fourth-order) case of the hierarchy. In particular, we present the generalized Lax pair and generalized soliton solutions, plane wave solutions, Akhmediev breathers, Kuznetsov-Ma breathers, periodic solutions, and rogue wave solutions for this infinite-order hierarchy. We find that "even- order" equations in the set affect phase and "stretching factors" in the solutions, while "odd-order" equations affect the velocities. Hence odd-order equation solutions can be real functions, while even-order equation solutions are always complex.

On a new class of completely integrable nonlinear wave equations. I. Infinitely many conservation laws

NASA Astrophysics Data System (ADS)

Nutku, Y.

1985-06-01

We point out a class of nonlinear wave equations which admit infinitely many conserved quantities. These equations are characterized by a pair of exact one-forms. The implication that they are closed gives rise to equations, the characteristics and Riemann invariants of which are readily obtained. The construction of the conservation laws requires the solution of a linear second-order equation which can be reduced to canonical form using the Riemann invariants. The hodograph transformation results in a similar linear equation. We discuss also the symplectic structure and Bäcklund transformations associated with these equations.

Gravitational waves — A review on the theoretical foundations of gravitational radiation

NASA Astrophysics Data System (ADS)

Dirkes, Alain

2018-05-01

In this paper, we review the theoretical foundations of gravitational waves in the framework of Albert Einstein’s theory of general relativity. Following Einstein’s early efforts, we first derive the linearized Einstein field equations and work out the corresponding gravitational wave equation. Moreover, we present the gravitational potentials in the far away wave zone field point approximation obtained from the relaxed Einstein field equations. We close this review by taking a closer look on the radiative losses of gravitating n-body systems and present some aspects of the current interferometric gravitational waves detectors. Each section has a separate appendix contribution where further computational details are displayed. To conclude, we summarize the main results and present a brief outlook in terms of current ongoing efforts to build a spaced-based gravitational wave observatory.

Application of wave mechanics theory to fluid dynamics problems: Fundamentals

NASA Technical Reports Server (NTRS)

Krzywoblocki, M. Z. V.

1974-01-01

The application of the basic formalistic elements of wave mechanics theory is discussed. The theory is used to describe the physical phenomena on the microscopic level, the fluid dynamics of gases and liquids, and the analysis of physical phenomena on the macroscopic (visually observable) level. The practical advantages of relating the two fields of wave mechanics and fluid mechanics through the use of the Schroedinger equation constitute the approach to this relationship. Some of the subjects include: (1) fundamental aspects of wave mechanics theory, (2) laminarity of flow, (3) velocity potential, (4) disturbances in fluids, (5) introductory elements of the bifurcation theory, and (6) physiological aspects in fluid dynamics.

Acoustic wave simulation using an overset grid for the global monitoring system

NASA Astrophysics Data System (ADS)

Kushida, N.; Le Bras, R.

2017-12-01

The International Monitoring System of the Comprehensive Nuclear-Test-Ban Treaty Organization (CTBTO) has been monitoring hydro-acoustic and infrasound waves over the globe. Because of the complex natures of the oceans and the atmosphere, computer simulation can play an important role in understanding the observed signals. In this regard, methods which depend on partial differential equations and require minimum modelling, are preferable. So far, to our best knowledge, acoustic wave propagation simulations based on partial differential equations on such a large scale have not been performed (pp 147 - 161 of ref [1], [2]). The main difficulties in building such simulation codes are: (1) considering the inhomogeneity of medium including background flows, (2) high aspect ratio of computational domain, (3) stability during long time integration. To overcome these difficulties, we employ a two-dimensional finite different (FDM) scheme on spherical coordinates with the Yin-Yang overset grid[3] solving the governing equation of acoustic waves introduces by Ostashev et. al.[4]. The comparison with real recording examples in hydro-acoustic will be presented at the conference. [1] Paul C. Etter: Underwater Acoustic Modeling and Simulation, Fourth Edition, CRC Press, 2013. [2] LIAN WANG et. al.: REVIEW OF UNDERWATER ACOUSTIC PROPAGATION MODELS, NPL Report AC 12, 2014. [3] A. Kageyama and T. Sato: "Yin-Yang grid": An overset grid in spherical geometry, Geochem. Geophys. Geosyst., 5, Q09005, 2004. [4] Vladimir E. Ostashev et. al: Equations for finite-difference, time-domain simulation of sound propagation in moving inhomogeneous media and numerical implementation, Acoustical Society of America. DOI: 10.1121/1.1841531, 2005.

Rogue periodic waves of the focusing nonlinear Schrödinger equation

NASA Astrophysics Data System (ADS)

Chen, Jinbing; Pelinovsky, Dmitry E.

2018-02-01

Rogue periodic waves stand for rogue waves on a periodic background. The nonlinear Schrödinger equation in the focusing case admits two families of periodic wave solutions expressed by the Jacobian elliptic functions dn and cn. Both periodic waves are modulationally unstable with respect to long-wave perturbations. Exact solutions for the rogue periodic waves are constructed by using the explicit expressions for the periodic eigenfunctions of the Zakharov-Shabat spectral problem and the Darboux transformations. These exact solutions generalize the classical rogue wave (the so-called Peregrine's breather). The magnification factor of the rogue periodic waves is computed as a function of the elliptic modulus. Rogue periodic waves constructed here are compared with the rogue wave patterns obtained numerically in recent publications.