An integrable shallow water equation with linear and nonlinear dispersion.
Dullin, H R; Gottwald, G A; Holm, D D
2001-11-01
We use asymptotic analysis and a near-identity normal form transformation from water wave theory to derive a 1+1 unidirectional nonlinear wave equation that combines the linear dispersion of the Korteweg-deVries (KdV) equation with the nonlinear/nonlocal dispersion of the Camassa-Holm (CH) equation. This equation is one order more accurate in asymptotic approximation beyond KdV, yet it still preserves complete integrability via the inverse scattering transform method. Its traveling wave solutions contain both the KdV solitons and the CH peakons as limiting cases. PMID:11690414
The zero dispersion limits of nonlinear wave equations
Tso, T.
1992-01-01
In chapter 2 the author uses functional analytic methods and conservation laws to solve the initial-value problem for the Korteweg-de Vries equation, the Benjamin-Bona-Mahony equation, and the nonlinear Schroedinger equation for initial data that satisfy some suitable conditions. In chapter 3 the energy estimates are used to show that the strong convergence of the family of the solutions of the KdV equation obtained in chapter 2 in H[sup 3](R) as [epsilon] [yields] 0; also, it is shown that the strong L[sup 2](R)-limit of the solutions of the BBM equation as [epsilon] [yields] 0 before a critical time. In chapter 4 the author uses the Whitham modulation theory and averaging method to find the 2[pi]-periodic solutions and the modulation equations of the KdV equation, the BBM equation, the Klein-Gordon equation, the NLS equation, the mKdV equation, and the P-system. It is shown that the modulation equations of the KdV equation, the K-G equation, the NLS equation, and the mKdV equation are hyperbolic but those of the BBM equation and the P-system are not hyperbolic. Also, the relations are studied of the KdV equation and the mKdV equation. Finally, the author studies the complex mKdV equation to compare with the NLS equation, and then study the complex gKdV equation.
Symmetry analysis for a class of nonlinear dispersive equations
NASA Astrophysics Data System (ADS)
Charalambous, K.; Sophocleous, C.
2015-05-01
A class of dispersive equations is studied within the framework of group analysis of differential equations. The enhanced Lie group classification is achieved. The complete list of equivalence transformations is presented. It is shown that certain equations from the class admit nonclassical reductions. Potential and potential nonclassical symmetries are also considered.
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.
On a hierarchy of nonlinearly dispersive generalized Korteweg - de Vries evolution equations
Christov, Ivan C.
2015-08-20
We propose a hierarchy of nonlinearly dispersive generalized Korteweg–de Vries (KdV) evolution equations based on a modification of the Lagrangian density whose induced action functional the KdV equation extremizes. Two recent nonlinear evolution equations describing wave propagation in certain generalized continua with an inherent material length scale are members of the proposed hierarchy. Like KdV, the equations from the proposed hierarchy possess Hamiltonian structure. Unlike KdV, the solutions to these equations can be compact (i.e., they vanish outside of some open interval) and, in addition, peaked. Implicit solutions for these peaked, compact traveling waves (“peakompactons”) are presented.
Construction of the wave operator for non-linear dispersive equations
NASA Astrophysics Data System (ADS)
Tsuruta, Kai Erik
In this thesis, we will study non-linear dispersive equations. The primary focus will be on the construction of the positive-time wave operator for such equations. The positive-time wave operator problem arises in the study of the asymptotics of a partial differential equation. It is a map from a space of initial data X into itself, and is loosely defined as follows: Suppose that for a solution ψlin to the dispersive equation with no non-linearity and initial data ψ +, there exists a unique solution ψ to the non-linear equation with initial data ψ0 such that ψ behaves as ψ lin as t → infinity. Then the wave operator is the map W+ that takes ψ + to ψ0. By its definition, W+ is injective. An important additional question is whether or not the map is also surjective. If so, then every non-linear solution emanating from X behaves, in some sense, linearly as it evolves (this is known as asymptotic completeness). Thus, there is some justification for treating these solutions as their much simpler linear counterparts. The main results presented in this thesis revolve around the construction of the wave operator(s) at critical non-linearities. We will study the "semi-relativistic" Schrodinger equation as well as the Klein-Gordon-Schrodinger system on R2 . In both cases, we will impose fairly general quadratic non-linearities for which conservation laws cannot be relied upon. These non-linearities fall below the scaling required to employ such tools as the Strichartz estimates. We instead adapt the "first iteration method" of Jang, Li, and Zhang to our setting which depends crucially on the critical decay of the non-linear interaction of the linear evolution. To see the critical decay in our problem, careful analysis is needed to treat the regime where one has spatial and/or time resonance.
Symmetries of the TDNLS equations for weakly nonlinear dispersive MHD waves
NASA Technical Reports Server (NTRS)
Webb, G. M.; Brio, M.; Zank, G. P.
1995-01-01
In this paper we consider the symmetries and conservation laws for the TDNLS equations derived by Hada (1993) and Brio, Hunter and Johnson, to describe the propagation of weakly nonlinear dispersive MHD waves in beta approximately 1 plasmas. The equations describe the interaction of the Alfven and magnetoacoustic modes near the triple umbilic, where the fast magnetosonic, slow magnetosonic and Alfven speeds coincide and a(g)(exp 2) = V(A)(exp 2) where a(g) is the gas sound speed and V(A) is the Alfven speed. We discuss Lagrangian and Hamiltonian formulations, and similarity solutions for the equations.
The modulational instability for the TDNLS equations for weakly nonlinear dispersive MHD waves
NASA Technical Reports Server (NTRS)
Webb, G. M.; Brio, M.; Zank, G. P.
1995-01-01
In this paper we study the modulational instability for the TDNLS equations derived by Hada (1993) and Brio, Hunter, and Johnson to describe the propagation of weakly nonlinear dispersive MHD waves in beta approximately 1 plasmas. We employ Whitham's averaged Lagrangian method to study the modulational instability. This complements studies of the modulational instability by Hada (1993) and Hollweg (1994), who did not use the averaged Lagrangian approach.
NASA Astrophysics Data System (ADS)
Mohammed, K. Elboree
2015-10-01
In this paper, we investigate the traveling wave solutions for the nonlinear dispersive equation, Korteweg-de Vries Zakharov-Kuznetsov (KdV-ZK) equation and complex coupled KdV system by using extended simplest equation method, and then derive the hyperbolic function solutions include soliton solutions, trigonometric function solutions include periodic solutions with special values for double parameters and rational solutions. The properties of such solutions are shown by figures. The results show that this method is an effective and a powerful tool for handling the solutions of nonlinear partial differential equations (NLEEs) in mathematical physics.
NASA Astrophysics Data System (ADS)
Kim, Bong-Sik
Three dimensional (3D) Navier-Stokes-alpha equations are considered for uniformly rotating geophysical fluid flows (large Coriolis parameter f = 2O). The Navier-Stokes-alpha equations are a nonlinear dispersive regularization of usual Navier-Stokes equations obtained by Lagrangian averaging. The focus is on the existence and global regularity of solutions of the 3D rotating Navier-Stokes-alpha equations and the uniform convergence of these solutions to those of the original 3D rotating Navier-Stokes equations for large Coriolis parameters f as alpha → 0. Methods are based on fast singular oscillating limits and results are obtained for periodic boundary conditions for all domain aspect ratios, including the case of three wave resonances which yields nonlinear "2½-dimensional" limit resonant equations for f → 0. The existence and global regularity of solutions of limit resonant equations is established, uniformly in alpha. Bootstrapping from global regularity of the limit equations, the existence of a regular solution of the full 3D rotating Navier-Stokes-alpha equations for large f for an infinite time is established. Then, the uniform convergence of a regular solution of the 3D rotating Navier-Stokes-alpha equations (alpha ≠ 0) to the one of the original 3D rotating NavierStokes equations (alpha = 0) for f large but fixed as alpha → 0 follows; this implies "shadowing" of trajectories of the limit dynamical systems by those of the perturbed alpha-dynamical systems. All the estimates are uniform in alpha, in contrast with previous estimates in the literature which blow up as alpha → 0. Finally, the existence of global attractors as well as exponential attractors is established for large f and the estimates are uniform in alpha.
NASA Astrophysics Data System (ADS)
Bona, J. L.; Chen, M.; Saut, J.-C.
2004-05-01
In part I of this work (Bona J L, Chen M and Saut J-C 2002 Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media I: Derivation and the linear theory J. Nonlinear Sci. 12 283-318), a four-parameter family of Boussinesq systems was derived to describe the propagation of surface water waves. Similar systems are expected to arise in other physical settings where the dominant aspects of propagation are a balance between the nonlinear effects of convection and the linear effects of frequency dispersion. In addition to deriving these systems, we determined in part I exactly which of them are linearly well posed in various natural function classes. It was argued that linear well-posedness is a natural necessary requirement for the possible physical relevance of the model in question. In this paper, it is shown that the first-order correct models that are linearly well posed are in fact locally nonlinearly well posed. Moreover, in certain specific cases, global well-posedness is established for physically relevant initial data. In part I, higher-order correct models were also derived. A preliminary analysis of a promising subclass of these models shows them to be well posed.
NASA Astrophysics Data System (ADS)
Zhang, Lijun; Chen, Li-Qun; Zhang, Jianming
2013-10-01
Bifurcation and exact solutions of the modified nonlinearly dispersive mK (m,n,k) equation with nonlinear dispersion um-1ut+a(un)x+b(uk)xxx = 0,nk≠0 are investigated in this paper. As a result, under different parameter conditions, abundant compactons, peakons and solitary solutions including not only some known results but also some new ones are obtained. We also point out the original reason of the existence of the non-smooth traveling wave solutions. The approach we used here is also suitable for the study of traveling wave solutions of some other nonlinear equations.
NASA Astrophysics Data System (ADS)
Li, Wan-Tong; Wang, Jia-Bing; Zhang, Li
2016-08-01
This paper is concerned with the new types of entire solutions other than traveling wave solutions of nonlocal dispersal equations with monostable nonlinearity in space periodic habitats. We first establish the existence and properties of spatially periodic solutions connecting two steady states. Then new types of entire solutions are constructed by combining the rightward and leftward pulsating traveling fronts with different speeds and a spatially periodic solution. Finally, for a class of special heterogeneous reaction, we further establish the uniqueness of entire solutions and the continuous dependence of such an entire solution on parameters, such as wave speeds and the shifted variables. In other words, we build a five-dimensional manifold of solutions and the traveling wave solutions are on the boundary of the manifold.
NASA Astrophysics Data System (ADS)
Charalampidis, E. G.; Kevrekidis, P. G.; Frantzeskakis, D. J.; Malomed, B. A.
2016-08-01
We consider a two-component, two-dimensional nonlinear Schrödinger system with unequal dispersion coefficients and self-defocusing nonlinearities, chiefly with equal strengths of the self- and cross-interactions. In this setting, a natural waveform with a nonvanishing background in one component is a vortex, which induces an effective potential well in the second component, via the nonlinear coupling of the two components. We show that the potential well may support not only the fundamental bound state, but also multiring excited radial state complexes for suitable ranges of values of the dispersion coefficient of the second component. We systematically explore the existence, stability, and nonlinear dynamics of these states. The complexes involving the excited radial states are weakly unstable, with a growth rate depending on the dispersion of the second component. Their evolution leads to transformation of the multiring complexes into stable vortex-bright solitons ones with the fundamental state in the second component. The excited states may be stabilized by a harmonic-oscillator trapping potential, as well as by unequal strengths of the self- and cross-repulsive nonlinearities.
Nonlinear modulation of periodic waves in the small dispersion limit of the Benjamin-Ono equation
NASA Astrophysics Data System (ADS)
Matsuno, Y.
1998-12-01
The Whitham modulation theory is used to construct large time asymptotic solutions of the Benjamin-Ono (BO) equation in the small dispersion limit. For a wide class of initial data, asymptotic solutions are represented by a single-phase periodic solution of the BO equation with slowly varying amplitude and wave number. The Whitham system of modulation equations for these wave parameters has a very simple structure, and it can be solved exactly under appropriate boundary conditions. It is found that the oscillating zone expands with time, and eventually evolves into a train of solitary waves. In the case of localized initial data, the number density function of solitary waves is derived in a closed form. The resulting expression coincides with the corresponding formula obtained from the asymptotic theory based on the conservation laws of the BO equation. For steplike initial data, the total number of created solitary waves increases without limit in proportion to time.
NASA Astrophysics Data System (ADS)
Arqub, Omar Abu; El-Ajou, Ahmad; Momani, Shaher
2015-07-01
Building fractional mathematical models for specific phenomena and developing numerical or analytical solutions for these fractional mathematical models are crucial issues in mathematics, physics, and engineering. In this work, a new analytical technique for constructing and predicting solitary pattern solutions of time-fractional dispersive partial differential equations is proposed based on the generalized Taylor series formula and residual error function. The new approach provides solutions in the form of a rapidly convergent series with easily computable components using symbolic computation software. For method evaluation and validation, the proposed technique was applied to three different models and compared with some of the well-known methods. The resultant simulations clearly demonstrate the superiority and potentiality of the proposed technique in terms of the quality performance and accuracy of substructure preservation in the construct, as well as the prediction of solitary pattern solutions for time-fractional dispersive partial differential equations.
NASA Astrophysics Data System (ADS)
Matsuno, Yoshimasa
2014-03-01
We develop a direct method for solving a modified Camassa-Holm equation with cubic nonlinearity and linear dispersion under the rapidly decreasing boundary condition. We obtain a compact parametric representation for the multisoliton solutions and investigate their properties. We show that the introduction of a linear dispersive term exhibits various new features in the structure of solutions. In particular, we find the smooth solitons whose characteristics are different from those of the Camassa-Holm equation, as well as the novel types of singular solitons. A remarkable feature of the soliton solutions is that the underlying structure of the associated tau-functions is the same as that of a model equation for shallow-water waves introduced by Ablowitz et al (1974 Stud. Appl. Math. 53 249-315). Finally, we demonstrate that the short-wave limit of the soliton solutions recovers the soliton solutions of the short pulse equation which describes the propagation of ultra-short optical pulses in nonlinear media.
Fujioka, J; Espinosa, A
2015-11-01
In this article, we show that if the nonlinear Schrödinger (NLS) equation is generalized by simultaneously taking into account higher-order dispersion, a quintic nonlinearity, and self-steepening terms, the resulting equation is interesting as it has exact soliton solutions which may be (depending on the values of the coefficients) stable or unstable, standard or "embedded," fixed or "moving" (i.e., solitons which advance along the retarded-time axis). We investigate the stability of these solitons by means of a modified version of the Vakhitov-Kolokolov criterion, and numerical tests are carried out to corroborate that these solitons respond differently to perturbations. It is shown that this generalized NLS equation can be derived from a Lagrangian density which contains an auxiliary variable, and Noether's theorem is then used to show that the invariance of the action integral under infinitesimal gauge transformations generates a whole family of conserved quantities. Finally, we study if this equation has the Painlevé property. PMID:26627574
NASA Astrophysics Data System (ADS)
Zhang, Guo-Bao; Ma, Ruyun
2014-10-01
This paper is concerned with the traveling wave solutions and the spreading speeds for a nonlocal dispersal equation with convolution-type crossing-monostable nonlinearity, which is motivated by an age-structured population model with time delay. We first prove the existence of traveling wave solution with critical wave speed c = c*. By introducing two auxiliary monotone birth functions and using a fluctuation method, we further show that the number c = c* is also the spreading speed of the corresponding initial value problem with compact support. Then, the nonexistence of traveling wave solutions for c < c* is established. Finally, by means of the (technical) weighted energy method, we prove that the traveling wave with large speed is exponentially stable, when the initial perturbation around the wave is relatively small in a weighted norm.
NASA Astrophysics Data System (ADS)
Beech, Robert; Osman, Frederick
2005-10-01
This paper will present the nonlinearity and dispersion effects involved in propagation of optical solitons, which can be understood by using a numerical routine to solve the nonlinear Schrödinger equation (NLSE). Here, Mathematica v5© (Wolfram, 2003) is used to explore in depth several features of optical solitons formation and propagation. These numerical routines were implemented through the use of Mathematica v5© and the results give a very clear idea of this interesting and important practical phenomenon. It is hoped that this work will open up an important new approach to the cause, effect, and correction of interference from secondary radiation found in the uses of soliton waves in lasers and in optical fiber telecommunication. It is believed that these results will be of considerable use in any work or research in this field and in self-focusing properties of the soliton (Osman et al., 2004a, 2004b; Hora, 1991). In a previous paper on this topic (Beech & Osman, 2004), it was shown that solitons of NLSE radiate. This paper goes on from there to show that these radiations only occur in solitons derived from cubic, or odd-numbered higher orders of NLSE, and that there are no such radiations from solitons of quadratic, or even-numbered higher order of NLSE. It is anticipated that this will stimulate research into practical means to control or eliminate such radiations.
NASA Astrophysics Data System (ADS)
Ramadan, Omar
2015-09-01
In this paper, systematic wave-equation finite difference time domain (WE-FDTD) formulations are presented for modeling electromagnetic wave-propagation in linear and nonlinear dispersive materials. In the proposed formulations, the complex conjugate pole residue (CCPR) pairs model is adopted in deriving a unified dispersive WE-FDTD algorithm that allows modeling different dispersive materials, such as Debye, Drude and Lorentz, in the same manner with the minimal additional auxiliary variables. Moreover, the proposed formulations are incorporated with the wave-equation perfectly matched layer (WE-PML) to construct a material independent mesh truncating technique that can be used for modeling general frequency-dependent open region problems. Several numerical examples involving linear and nonlinear dispersive materials are included to show the validity of the proposed formulations.
Chen, Yong; Yan, Zhenya
2016-01-01
Solitons are of the important significant in many fields of nonlinear science such as nonlinear optics, Bose-Einstein condensates, plamas physics, biology, fluid mechanics, and etc. The stable solitons have been captured not only theoretically and experimentally in both linear and nonlinear Schrödinger (NLS) equations in the presence of non-Hermitian potentials since the concept of the parity-time -symmetry was introduced in 1998. In this paper, we present novel bright solitons of the NLS equation with third-order dispersion in some complex -symmetric potentials (e.g., physically relevant -symmetric Scarff-II-like and harmonic-Gaussian potentials). We find stable nonlinear modes even if the respective linear -symmetric phases are broken. Moreover, we also use the adiabatic changes of the control parameters to excite the initial modes related to exact solitons to reach stable nonlinear modes. The elastic interactions of two solitons are exhibited in the third-order NLS equation with -symmetric potentials. Our results predict the dynamical phenomena of soliton equations in the presence of third-order dispersion and -symmetric potentials arising in nonlinear fiber optics and other physically relevant fields. PMID:27002543
Chen, Yong; Yan, Zhenya
2016-01-01
Solitons are of the important significant in many fields of nonlinear science such as nonlinear optics, Bose-Einstein condensates, plamas physics, biology, fluid mechanics, and etc. The stable solitons have been captured not only theoretically and experimentally in both linear and nonlinear Schrödinger (NLS) equations in the presence of non-Hermitian potentials since the concept of the parity-time -symmetry was introduced in 1998. In this paper, we present novel bright solitons of the NLS equation with third-order dispersion in some complex -symmetric potentials (e.g., physically relevant -symmetric Scarff-II-like and harmonic-Gaussian potentials). We find stable nonlinear modes even if the respective linear -symmetric phases are broken. Moreover, we also use the adiabatic changes of the control parameters to excite the initial modes related to exact solitons to reach stable nonlinear modes. The elastic interactions of two solitons are exhibited in the third-order NLS equation with -symmetric potentials. Our results predict the dynamical phenomena of soliton equations in the presence of third-order dispersion and -symmetric potentials arising in nonlinear fiber optics and other physically relevant fields. PMID:27002543
NASA Astrophysics Data System (ADS)
Chen, Yong; Yan, Zhenya
2016-03-01
Solitons are of the important significant in many fields of nonlinear science such as nonlinear optics, Bose-Einstein condensates, plamas physics, biology, fluid mechanics, and etc. The stable solitons have been captured not only theoretically and experimentally in both linear and nonlinear Schrödinger (NLS) equations in the presence of non-Hermitian potentials since the concept of the parity-time -symmetry was introduced in 1998. In this paper, we present novel bright solitons of the NLS equation with third-order dispersion in some complex -symmetric potentials (e.g., physically relevant -symmetric Scarff-II-like and harmonic-Gaussian potentials). We find stable nonlinear modes even if the respective linear -symmetric phases are broken. Moreover, we also use the adiabatic changes of the control parameters to excite the initial modes related to exact solitons to reach stable nonlinear modes. The elastic interactions of two solitons are exhibited in the third-order NLS equation with -symmetric potentials. Our results predict the dynamical phenomena of soliton equations in the presence of third-order dispersion and -symmetric potentials arising in nonlinear fiber optics and other physically relevant fields.
Nonlinear ordinary difference equations
NASA Technical Reports Server (NTRS)
Caughey, T. K.
1979-01-01
Future space vehicles will be relatively large and flexible, and active control will be necessary to maintain geometrical configuration. While the stresses and strains in these space vehicles are not expected to be excessively large, their cumulative effects will cause significant geometrical nonlinearities to appear in the equations of motion, in addition to the nonlinearities caused by material properties. Since the only effective tool for the analysis of such large complex structures is the digital computer, it will be necessary to gain a better understanding of the nonlinear ordinary difference equations which result from the time discretization of the semidiscrete equations of motion for such structures.
Nikolaevskiy equation with dispersion.
Simbawa, Eman; Matthews, Paul C; Cox, Stephen M
2010-03-01
The Nikolaevskiy equation was originally proposed as a model for seismic waves and is also a model for a wide variety of systems incorporating a neutral "Goldstone" mode, including electroconvection and reaction-diffusion systems. It is known to exhibit chaotic dynamics at the onset of pattern formation, at least when the dispersive terms in the equation are suppressed, as is commonly the practice in previous analyses. In this paper, the effects of reinstating the dispersive terms are examined. It is shown that such terms can stabilize some of the spatially periodic traveling waves; this allows us to study the loss of stability and transition to chaos of the waves. The secondary stability diagram ("Busse balloon") for the traveling waves can be remarkably complicated. PMID:20365845
Scalora, Michael; Syrchin, Maxim S; Akozbek, Neset; Poliakov, Evgeni Y; D'Aguanno, Giuseppe; Mattiucci, Nadia; Bloemer, Mark J; Zheltikov, Aleksei M
2005-07-01
A new generalized nonlinear Schrödinger equation describing the propagation of ultrashort pulses in bulk media exhibiting frequency dependent dielectric susceptibility and magnetic permeability is derived and used to characterize wave propagation in a negative index material. The equation has new features that are distinct from ordinary materials (mu=1): the linear and nonlinear coefficients can be tailored through the linear properties of the medium to attain any combination of signs unachievable in ordinary matter, with significant potential to realize a wide class of solitary waves. PMID:16090616
NASA Technical Reports Server (NTRS)
Goorjian, Peter M.; Taflove, Allen
1992-01-01
The initial results for femtosecond electromagnetic soliton propagation and collision obtained from first principles, i.e., by a direct time integration of Maxwell's equations are reported. 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 the modeling of 2D and 3D optical soliton propagation, scattering, and switching from the full-vector Maxwell's equations.
Nonlinear gyrokinetic equations
Dubin, D.H.E.; Krommes, J.A.; Oberman, C.; Lee, W.W.
1983-03-01
Nonlinear gyrokinetic equations are derived from a systematic Hamiltonian theory. The derivation employs Lie transforms and a noncanonical perturbation theory first used by Littlejohn for the simpler problem of asymptotically small gyroradius. For definiteness, we emphasize the limit of electrostatic fluctuations in slab geometry; however, there is a straight-forward generalization to arbitrary field geometry and electromagnetic perturbations. An energy invariant for the nonlinear system is derived, and various of its limits are considered. The weak turbulence theory of the equations is examined. In particular, the wave kinetic equation of Galeev and Sagdeev is derived from an asystematic truncation of the equations, implying that this equation fails to consider all gyrokinetic effects. The equations are simplified for the case of small but finite gyroradius and put in a form suitable for efficient computer simulation. Although it is possible to derive the Terry-Horton and Hasegawa-Mima equations as limiting cases of our theory, several new nonlinear terms absent from conventional theories appear and are discussed.
Nonlinear differential equations
Dresner, L.
1988-01-01
This report is the text of a graduate course on nonlinear differential equations given by the author at the University of Wisconsin-Madison during the summer of 1987. The topics covered are: direction fields of first-order differential equations; the Lie (group) theory of ordinary differential equations; similarity solutions of second-order partial differential equations; maximum principles and differential inequalities; monotone operators and iteration; complementary variational principles; and stability of numerical methods. The report should be of interest to graduate students, faculty, and practicing scientists and engineers. No prior knowledge is required beyond a good working knowledge of the calculus. The emphasis is on practical results. Most of the illustrative examples are taken from the fields of nonlinear diffusion, heat and mass transfer, applied superconductivity, and helium cryogenics.
Dispersion of nonlinearity and modulation instability in subwavelength semiconductor waveguides.
Gorbach, A V; Zhao, X; Skryabin, D V
2011-05-01
Tight confinement of light in subwavelength waveguides induces substantial dispersion of their nonlinear response. We demonstrate that this dispersion of nonlinearity can lead to the modulational instability in the regime of normal group velocity dispersion through the mechanism independent from higher order dispersions of linear waves. A simple phenomenological model describing this effect is the nonlinear Schrödinger equation with the intensity dependent group velocity dispersion. PMID:21643190
Engineering integrable nonautonomous nonlinear Schroedinger equations
He Xugang; Zhao Dun; Li Lin; Luo Honggang
2009-05-15
We investigate Painleve integrability of a generalized nonautonomous one-dimensional nonlinear Schroedinger (NLS) equation with time- and space-dependent dispersion, nonlinearity, and external potentials. Through the Painleve analysis some explicit requirements on the dispersion, nonlinearity, dissipation/gain, and the external potential as well as the constraint conditions are identified. It provides an explicit way to engineer integrable nonautonomous NLS equations at least in the sense of Painleve integrability. Furthermore analytical solutions of this class of integrable nonautonomous NLS equations can be obtained explicitly from the solutions of the standard NLS equation by a general transformation. The result provides a significant way to control coherently the soliton dynamics in the corresponding nonlinear systems, as that in Bose-Einstein condensate experiments. We analyze explicitly the soliton dynamics under the nonlinearity management and the external potentials and discuss its application in the matter-wave dynamics. Some comparisons with the previous works have also been discussed.
NASA Astrophysics Data System (ADS)
Bona, G.; Chen, J. A.; Saut, Jing Ping
2002-08-01
Considered herein are a number of variants of the classical Boussinesq system and their higher-order generalizations. Such equations were first derived by Boussinesq to describe the two-way propagation of small-amplitude, long wavelength, gravity waves on the surface of water in a canal. These systems arise also when modeling the propagation of long-crested waves on large lakes or the ocean and in other contexts. Depending on the modeling of dispersion, the resulting system may or may not have a linearization about the rest state which is well posed. Even when well posed, the linearized system may exhibit a lack of conservation of energy that is at odds with its status as an approximation to the Euler equations. In the present script, we derive a four-parameter family of Boussinesq systems from the two-dimensional Euler equations for free-surface flow and formulate criteria to help decide which of these equations one might choose in a given modeling situation. The analysis of the systems according to these criteria is initiated.
Dispersion in tidally averaged transport equation
Cheng, R.T.; Casulli, V.
1992-01-01
A general governing inter-tidal transport equation for conservative solutes has been derived without invoking the weakly nonlinear approximation. The governing inter-tidal transport equation is a convection-dispersion equation in which the convective velocity is a mean Lagrangian residual current, and the inter-tidal dispersion coefficient is defined by a dispersion patch. When the weakly nonlinear condition is violated, the physical significance of the Stokes' drift, as used in tidal dynamics, becomes questionable. For nonlinear problems, analytical solutions for the mean Lagrangian residual current and for the inter-tidal dispersion coefficient do not exist, they must be determined numerically. A rectangular tidal inlet with a constriction is used in the first example. The solutions of the residual currents and the computed properties of the inter-tidal dispersion coefficient are used to illuminate the mechanisms of the inter-tidal transport processes. Then, the present formulation is tested in a geometrically complex tidal estuary – San Francisco Bay, California. The computed inter-tidal dispersion coefficients are in the range between 5×104 and 5×106 cm2/sec., which are consistent with the values reported in the literature
Solving Nonlinear Coupled Differential Equations
NASA Technical Reports Server (NTRS)
Mitchell, L.; David, J.
1986-01-01
Harmonic balance method developed to obtain approximate steady-state solutions for nonlinear coupled ordinary differential equations. Method usable with transfer matrices commonly used to analyze shaft systems. Solution to nonlinear equation, with periodic forcing function represented as sum of series similar to Fourier series but with form of terms suggested by equation itself.
Spatiotemporal coupling in dispersive nonlinear planar waveguides
NASA Astrophysics Data System (ADS)
Ryan, Andrew T.; Agrawal, Govind P.
1995-12-01
The multidimensional nonlinear Schrodinger equation governs the spatial and temporal evolution of an optical field inside a nonlinear dispersive medium. Although spatial (diffractive) and temporal (dispersive) effects can be studied independently in a linear medium, they become mutually coupled in a nonlinear medium. We present the results of numerical simulations showing this spatiotemporal coupling for ultrashort pulses propagating in dispersive Kerr media. We investigate how spatiotemporal coupling affects the behavior of the optical field in each of the four regimes defined by the type of group-velocity dispersion (normal or anomalous) and the type of nonlinearity (focusing or defocusing). We show that dispersion, through spatiotemporal coupling, can either enhance or suppress self-focusing and self-defocusing. Similarly, we demonstrate that diffraction can either enhance or suppress pulse compression or broadening. We also discuss how these effects can be controlled with optical phase modulation, such as that provided by a lens (spatial phase modulation) or frequency chirping (temporal phase modulation). Copyright (c) 1995 Optical Society of America
Solitons and nonlinear wave equations
Dodd, Roger K.; Eilbeck, J. Chris; Gibbon, John D.; Morris, Hedley C.
1982-01-01
A discussion of the theory and applications of classical solitons is presented with a brief treatment of quantum mechanical effects which occur in particle physics and quantum field theory. The subjects addressed include: solitary waves and solitons, scattering transforms, the Schroedinger equation and the Korteweg-de Vries equation, and the inverse method for the isospectral Schroedinger equation and the general solution of the solvable nonlinear equations. Also considered are: isolation of the Korteweg-de Vries equation in some physical examples, the Zakharov-Shabat/AKNS inverse method, kinks and the sine-Gordon equation, the nonlinear Schroedinger equation and wave resonance interactions, amplitude equations in unstable systems, and numerical studies of solitons. 45 references.
NASA Astrophysics Data System (ADS)
Ganapathy, R.; Kuriakose, V. C.
2002-04-01
We obtain conditions for the occurrence of cross-phase modulational instability in the normal dispersion regime for the coupled higher order nonlinear Schrödinger equation with higher order dispersion and nonlinear terms.
Absorbing Boundary Conditions For Optical Pulses In Dispersive, Nonlinear Materials
NASA Technical Reports Server (NTRS)
Goorjian, Peter M.; Kwak, Dochan (Technical Monitor)
1995-01-01
This paper will present results in computational nonlinear optics. An algorithm will be described that provides absorbing boundary conditions for optical pulses in dispersive, nonlinear materials. A new numerical absorber at the boundaries has been developed that is responsive to the spectral content of the pulse. 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. Comparisons will be shown of calculations that use the standard boundary conditions and the new ones.
Duffing's Equation and Nonlinear Resonance
ERIC Educational Resources Information Center
Fay, Temple H.
2003-01-01
The phenomenon of nonlinear resonance (sometimes called the "jump phenomenon") is examined and second-order van der Pol plane analysis is employed to indicate that this phenomenon is not a feature of the equation, but rather the result of accumulated round-off error, truncation error and algorithm error that distorts the true bounded solution onto…
Manikandan, K; Senthilvelan, M
2016-07-01
We construct spatiotemporal localized envelope solutions of a (3 + 1)-dimensional nonlinear Schrödinger equation with varying coefficients such as dispersion, nonlinearity and gain parameters through similarity transformation technique. The obtained localized rational solutions can serve as prototypes of rogue waves in different branches of science. We investigate the characteristics of constructed localized solutions in detail when it propagates through six different dispersion profiles, namely, constant, linear, Gaussian, hyperbolic, logarithm, and exponential. We also obtain expressions for the hump and valleys of rogue wave intensity profiles for these six dispersion profiles and study the trajectory of it in each case. Further, we analyze how the intensity of another localized solution, namely, breather, changes when it propagates through the aforementioned six dispersion profiles. Our studies reveal that these localized solutions co-exist with the collapsing solutions which are already found in the (3 + 1)-dimensional nonlinear Schrödinger equation. The obtained results will help to understand the corresponding localized wave phenomena in related fields. PMID:27475076
Linear superposition in nonlinear equations.
Khare, Avinash; Sukhatme, Uday
2002-06-17
Several nonlinear systems such as the Korteweg-de Vries (KdV) and modified KdV equations and lambda phi(4) theory possess periodic traveling wave solutions involving Jacobi elliptic functions. We show that suitable linear combinations of these known periodic solutions yield many additional solutions with different periods and velocities. This linear superposition procedure works by virtue of some remarkable new identities involving elliptic functions. PMID:12059300
Cauchy's dispersion equation reconsidered : dispersion in silicate glasses.
Smith, D. Y.; Inokuti, M.; Karstens, W.; Physics; Univ. of Vermont; St. Michael's College
2002-01-01
We formulate a novel method of characterizing optically transparent substances using dispersion theory. The refractive index is given by a generalized Cauchy dispersion equation with coefficients that are moments of the uv and ir absorptions. Mean dispersion, Abbe number, and partial dispersion are combinations of these moments. The empirical relation between index and dispersion for families of glasses appears as a consequence of Beer's law applied to the uv spectra.
Nonlinear SCHRÖDINGER-PAULI Equations
NASA Astrophysics Data System (ADS)
Ng, Wei Khim; Parwani, Rajesh R.
2011-11-01
We obtain novel nonlinear Schrüdinger-Pauli equations through a formal non-relativistic limit of appropriately constructed nonlinear Dirac equations. This procedure automatically provides a physical regularisation of potential singularities brought forward by the nonlinear terms and suggests how to regularise previous equations studied in the literature. The enhancement of contributions coming from the regularised singularities suggests that the obtained equations might be useful for future precision tests of quantum nonlinearity.
Tu, Haohua; Liu, Yuan; Lægsgaard, Jesper; Sharma, Utkarsh; Siegel, Martin; Kopf, Daniel; Boppart, Stephen A
2010-12-20
We quantitatively predict the observed continuum-like spectral broadening in a 90-mm weakly birefringent all-normal dispersion-flattened photonic crystal fiber pumped by 1041-nm 229-fs 76-MHz pulses from a solid-state Yb:KYW laser. The well-characterized continuum pulses span a bandwidth of up to 300 nm around the laser wavelength, allowing high spectral power density pulse shaping useful for various coherent control applications. We also identify the nonlinear polarization effect that limits the bandwidth of these continuum pulses, and therefore report the path toward a series of attractive alternative broadband coherent optical sources. PMID:21197060
Tu, Haohua; Liu, Yuan; Lægsgaard, Jesper; Sharma, Utkarsh; Siegel, Martin; Kopf, Daniel; Boppart, Stephen A.
2010-01-01
We quantitatively predict the observed continuum-like spectral broadening in a 90-mm weakly birefringent all-normal dispersion-flattened photonic crystal fiber pumped by 1041-nm 229-fs 76-MHz pulses from a solid-state Yb:KYW laser. The well-characterized continuum pulses span a bandwidth of up to 300 nm around the laser wavelength, allowing high spectral power density pulse shaping useful for various coherent control applications. We also identify the nonlinear polarization effect that limits the bandwidth of these continuum pulses, and therefore report the path toward a series of attractive alternative broadband coherent optical sources. PMID:21197060
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.
Computational studies of nonlinear dispersive plasma systems
NASA Astrophysics Data System (ADS)
Qian, Xin
Plasma systems with dispersive waves are ubiquitous. Dispersive waves have the property that their wave velocity depends on the wave number of the wave. These waves show up in weakly as well as strongly coupled plasmas, and play a significant role in the underlying plasma dynamics. Dispersive waves bring new challenges to the computer simulation of nonlinear phenomena. The goal of this thesis is to discuss two computational studies of plasma phenomena, one drawn from strongly coupled complex or dusty plasmas, and the other from weakly coupled hydrogen plasmas. In the realm of dusty plasmas, we focus on the problem of three-dimensional (3D) Mach cones which we study by means of Molecular Dynamics (MD) simulations, assuming that the dust particles interact via a Yukawa potential. While laboratory and MD simulations have explored thoroughly the properties of Mach cones in 2D, elucidating the important role of dispersive waves in the formation of multiple cones, the simulations presented in this thesis represent the first 3D MD studies of Mach cones in strongly coupled dusty plasmas. These results have qualitative similarities with experimental observations on 3D Mach cones from the PK-3 plus project, which studies complex plasmas under microgravity conditions aboard the International Space station. In the realm of weakly coupled plasmas, we present results on the application of non-oscillatory central schemes to Hall MHD reconnection problems, in which the presence of dispersive whistler waves presents a formidable challenge for numerical algorithms that rely on explicit time-stepping schemes. In particular, we focus on the semi-discrete central formulation of Kurganov and Tadmor (2000), which has the advantage that it allow for larger time steps, and with significantly smaller numerical viscosity, than fully discrete schemes. We implement the Hall MHD equations through the CentPACK software package that implements the Kurganov-Tadmor formulation for a wide range of
Acoustic nonlinearity in dispersive solids
NASA Technical Reports Server (NTRS)
Cantrell, John H.; Yost, William T.
1991-01-01
An investigation to consider the effects of dispersion on the generation of the static acoustic wave component is presented. It is considered that an acoustic toneburst may be modeled as a modulated continuous waveform and that the generated initial static displacement pulse may be viewed as a modulation-confined disturbance. A theoretical model for the generation of the acoustic modulation solitons evolved is developed and experimental evidence in samples of vitreous silica demonstrating the essential validity of the model is provided.
Nonlinear rheology of colloidal dispersions.
Brader, J M
2010-09-15
Colloidal dispersions are commonly encountered in everyday life and represent an important class of complex fluid. Of particular significance for many commercial products and industrial processes is the ability to control and manipulate the macroscopic flow response of a dispersion by tuning the microscopic interactions between the constituents. An important step towards attaining this goal is the development of robust theoretical methods for predicting from first-principles the rheology and nonequilibrium microstructure of well defined model systems subject to external flow. In this review we give an overview of some promising theoretical approaches and the phenomena they seek to describe, focusing, for simplicity, on systems for which the colloidal particles interact via strongly repulsive, spherically symmetric interactions. In presenting the various theories, we will consider first low volume fraction systems, for which a number of exact results may be derived, before moving on to consider the intermediate and high volume fraction states which present both the most interesting physics and the most demanding technical challenges. In the high volume fraction regime particular emphasis will be given to the rheology of dynamically arrested states. PMID:21386516
Spurious Solutions Of Nonlinear Differential Equations
NASA Technical Reports Server (NTRS)
Yee, H. C.; Sweby, P. K.; Griffiths, D. F.
1992-01-01
Report utilizes nonlinear-dynamics approach to investigate possible sources of errors and slow convergence and non-convergence of steady-state numerical solutions when using time-dependent approach for problems containing nonlinear source terms. Emphasizes implications for development of algorithms in CFD and computational sciences in general. Main fundamental conclusion of study is that qualitative features of nonlinear differential equations cannot be adequately represented by finite-difference method and vice versa.
Viscous Fluid Conduits as a Prototypical Nonlinear Dispersive Wave Platform
NASA Astrophysics Data System (ADS)
Lowman, Nicholas K.
This thesis is devoted to the comprehensive characterization of slowly modulated, nonlinear waves in dispersive media for physically-relevant systems using a threefold approach: analytical, long-time asymptotics, careful numerical simulations, and quantitative laboratory experiments. In particular, we use this interdisciplinary approach to establish a two-fluid, interfacial fluid flow setting known as viscous fluid conduits as an ideal platform for the experimental study of truly one dimensional, unidirectional solitary waves and dispersively regularized shock waves (DSWs). Starting from the full set of fluid equations for mass and linear momentum conservation, we use a multiple-scales, perturbation approach to derive a scalar, nonlinear, dispersive wave equation for the leading order interfacial dynamics of the system. Using a generalized form of the approximate model equation, we use numerical simulations and an analytical, nonlinear wave averaging technique, Whitham-El modulation theory, to derive the key physical features of interacting large amplitude solitary waves and DSWs. We then present the results of quantitative, experimental investigations into large amplitude solitary wave interactions and DSWs. Overtaking interactions of large amplitude solitary waves are shown to exhibit nearly elastic collisions and universal interaction geometries according to the Lax categories for KdV solitons, and to be in excellent agreement with the dynamics described by the approximate asymptotic model. The dispersive shock wave experiments presented here represent the most extensive comparison to date between theory and data of the key wavetrain parameters predicted by modulation theory. We observe strong agreement. Based on the work in this thesis, viscous fluid conduits provide a well-understood, controlled, table-top environment in which to study universal properties of dispersive hydrodynamics. Motivated by the study of wave propagation in the conduit system, we
Nonlinear quantum equations: Classical field theory
Rego-Monteiro, M. A.; Nobre, F. D.
2013-10-15
An exact classical field theory for nonlinear quantum equations is presented herein. It has been applied recently to a nonlinear Schrödinger equation, and it is shown herein to hold also for a nonlinear generalization of the Klein-Gordon equation. These generalizations were carried by introducing nonlinear terms, characterized by exponents depending on an index q, in such a way that the standard, linear equations, are recovered in the limit q→ 1. The main characteristic of this field theory consists on the fact that besides the usual Ψ(x(vector sign),t), a new field Φ(x(vector sign),t) needs to be introduced in the Lagrangian, as well. The field Φ(x(vector sign),t), which is defined by means of an additional equation, becomes Ψ{sup *}(x(vector sign),t) only when q→ 1. The solutions for the fields Ψ(x(vector sign),t) and Φ(x(vector sign),t) are found herein, being expressed in terms of a q-plane wave; moreover, both field equations lead to the relation E{sup 2}=p{sup 2}c{sup 2}+m{sup 2}c{sup 4}, for all values of q. The fact that such a classical field theory works well for two very distinct nonlinear quantum equations, namely, the Schrödinger and Klein-Gordon ones, suggests that this procedure should be appropriate for a wider class nonlinear equations. It is shown that the standard global gauge invariance is broken as a consequence of the nonlinearity.
Extended Trial Equation Method for Nonlinear Partial Differential Equations
NASA Astrophysics Data System (ADS)
Gepreel, Khaled A.; Nofal, Taher A.
2015-04-01
The main objective of this paper is to use the extended trial equation method to construct a series of some new solutions for some nonlinear partial differential equations (PDEs) in mathematical physics. We will construct the solutions in many different functions such as hyperbolic function solutions, trigonometric function solutions, Jacobi elliptic function solutions, and rational functional solutions for the nonlinear PDEs when the balance number is a real number via the Zhiber-Shabat nonlinear differential equation. The balance number of this method is not constant as we shown in other methods, but it is changed by changing the trial equation derivative definition. This method allowed us to construct many new types of solutions. It is shown by using the Maple software package that all obtained solutions satisfy the original PDEs.
Chromatic dispersions in highly nonlinear glass nanofibers
NASA Astrophysics Data System (ADS)
Chaudhari, Chitrarekha; Suzuki, Takenobu; Ohishi, Yasutake
2008-08-01
We design air cladding tellurite (TeO2), bismuth oxide (Bi2O3) based, and chalcogenide (As2S3) nanofibers, and calculate the chromatic dispersions. For each material, wavelength dependent propagation constants of the nanofiber are obtained from the exact solutions of the Maxwell's equations, and from the propagation constants the chromatic dispersion is calculated. We tailor the dispersion to zero at the communication wavelength, 1.5 μm, by proper selection of the core diameter of the nanofiber for all the above materials. We further explain the technique for flattening the zero dispersion in telecommunication window, using glass instead of air, as the cladding of the nanofiber structure. Using the glass cladding has the advantage of easy handling, specially, for the communication purposes. Further, the glass cladding causes larger effective index difference between various modes of the nanofiber, thus reducing the mode coupling. We present the numerical results of the dispersion flattening technique by assuming the borosilicate glass cladding to the chalcogenide As2S3 glass core nanofiber. With the borosilicate cladding the dispersion characteristics of the nanofiber change drastically and flattening of the zero dispersion is achieved at 1.408 μm wavelength, when the core diameter is 724 nm.
Evolution equation for non-linear cosmological perturbations
Brustein, Ram; Riotto, Antonio E-mail: Antonio.Riotto@cern.ch
2011-11-01
We present a novel approach, based entirely on the gravitational potential, for studying the evolution of non-linear cosmological matter perturbations. Starting from the perturbed Einstein equations, we integrate out the non-relativistic degrees of freedom of the cosmic fluid and obtain a single closed equation for the gravitational potential. We then verify the validity of the new equation by comparing its approximate solutions to known results in the theory of non-linear cosmological perturbations. First, we show explicitly that the perturbative solution of our equation matches the standard perturbative solutions. Next, using the mean field approximation to the equation, we show that its solution reproduces in a simple way the exponential suppression of the non-linear propagator on small scales due to the velocity dispersion. Our approach can therefore reproduce the main features of the renormalized perturbation theory and (time)-renormalization group approaches to the study of non-linear cosmological perturbations, with some possibly important differences. We conclude by a preliminary discussion of the nature of the full solutions of the equation and their significance.
Numerical study of fractional nonlinear Schrödinger equations.
Klein, Christian; Sparber, Christof; Markowich, Peter
2014-12-01
Using a Fourier spectral method, we provide a detailed numerical investigation of dispersive Schrödinger-type equations involving a fractional Laplacian in an one-dimensional case. By an appropriate choice of the dispersive exponent, both mass and energy sub- and supercritical regimes can be identified. This allows us to study the possibility of finite time blow-up versus global existence, the nature of the blow-up, the stability and instability of nonlinear ground states and the long-time dynamics of solutions. The latter is also studied in a semiclassical setting. Moreover, we numerically construct ground state solutions of the fractional nonlinear Schrödinger equation. PMID:25484604
Numerical study of fractional nonlinear Schrödinger equations
Klein, Christian; Sparber, Christof; Markowich, Peter
2014-01-01
Using a Fourier spectral method, we provide a detailed numerical investigation of dispersive Schrödinger-type equations involving a fractional Laplacian in an one-dimensional case. By an appropriate choice of the dispersive exponent, both mass and energy sub- and supercritical regimes can be identified. This allows us to study the possibility of finite time blow-up versus global existence, the nature of the blow-up, the stability and instability of nonlinear ground states and the long-time dynamics of solutions. The latter is also studied in a semiclassical setting. Moreover, we numerically construct ground state solutions of the fractional nonlinear Schrödinger equation. PMID:25484604
On a class of nonlinear dispersive-dissipative interactions
Rosenau, P.
1997-07-29
The authors study the prototypical, genuinely nonlinear, equation; u{sub t} + a(u{sup m}){sub x} + (u{sup n}){sub xxx} = {mu}(u{sup k}){sub xx}, a, {mu} = consts., which encompasses a wide variety of dissipative-dispersive interactions. The parametric surface k = (m + n)/2 separates diffusion dominated from dissipation dominated phenomena. On this surface dissipative and dispersive effects are in detailed balance for all amplitudes. In particular, the m = n + 2 = k + 1 subclass can be transformed into a form free of convection and dissipation making it accessible to theoretical studies. Both bounded and unbounded oscillations are found and certain exact solutions are presented. When a = (2{mu}3/){sup 2} the map yields a linear equation; rational, periodic and aperiodic solutions are constructed.
Modified non-linear Burgers' equations and cosmic ray shocks
NASA Technical Reports Server (NTRS)
Zank, G. P.; Webb, G. M.; Mckenzie, J. F.
1988-01-01
A reductive perturbation scheme is used to derive a generalized non-linear Burgers' equation, which includes the effects of dispersion, in the long wavelength regime for the two-fluid hydrodynamical model used to describe cosmic ray acceleration by the first-order Fermi process in astrophysical shocks. The generalized Burger's equation is derived for both relativistic and non-relativistic cosmic ray shocks, and describes the time evolution of weak shocks in the theory of diffusive shock acceleration. The inclusion of dispersive effects modifies the phase velocity of the shock obtained from the lower order non-linear Burger's equation through the introduction of higher order terms from the long wavelength dispersion equation. The travelling wave solution of the generalized Burgers' equation for a single shock shows that larger cosmic ray pressures result in broader shock transitions. The results for relativistic shocks show a steepening of the shock as the shock speed approaches the relativistic cosmic ray sound speed. The dependence of the shock speed on the cosmic ray pressure is also discussed.
Markovian master equation for nonlinear systems
NASA Astrophysics Data System (ADS)
de los Santos-Sánchez, O.; Récamier, J.; Jáuregui, R.
2015-06-01
Within the f-deformed oscillator formalism, we derive a Markovian master equation for the description of the damped dynamics of nonlinear systems that interact with their environment. The applicability of this treatment to the particular case of a Morse-like oscillator interacting with a thermal field is illustrated, and the decay of quantum coherence in such a system is analyzed in terms of the evolution on phase space of its nonlinear coherent states via the Wigner function.
Langevin equation model of dispersion in the convective boundary layer
Nasstrom, J S
1998-08-01
This dissertation presents the development and evaluation of a Lagrangian stochastic model of vertical dispersion of trace material in the convective boundary layer (CBL). This model is based on a Langevin equation of motion for a fluid particle, and assumes the fluid vertical velocity probability distribution is skewed and spatially homogeneous. This approach can account for the effect of large-scale, long-lived turbulent structures and skewed vertical velocity distributions found in the CBL. The form of the Langevin equation used has a linear (in velocity) deterministic acceleration and a skewed randomacceleration. For the case of homogeneous fluid velocity statistics, this ""linear-skewed" Langevin equation can be integrated explicitly, resulting in a relatively efficient numerical simulation method. It is shown that this approach is more efficient than an alternative using a "nonlinear-Gaussian" Langevin equation (with a nonlinear deterministic acceleration and a Gaussian random acceleration) assuming homogeneous turbulence, and much more efficient than alternative approaches using Langevin equation models assuming inhomogeneous turbulence. "Reflection" boundary conditions for selecting a new velocity for a particle that encounters a boundary at the top or bottom of the CBL were investigated. These include one method using the standard assumption that the magnitudes of the particle incident and reflected velocities are positively correlated, and two alternatives in which the magnitudes of these velocities are negatively correlated and uncorrelated. The constraint that spatial and velocity distributions of a well-mixed tracer must be the same as those of the fluid, was used to develop the Langevin equation models and the reflection boundary conditions. The two Langevin equation models and three reflection methods were successfully tested using cases for which exact, analytic statistical properties of particle velocity and position are known, including well
Algorithms For Integrating Nonlinear Differential Equations
NASA Technical Reports Server (NTRS)
Freed, A. D.; Walker, K. P.
1994-01-01
Improved algorithms developed for use in numerical integration of systems of nonhomogenous, nonlinear, first-order, ordinary differential equations. In comparison with integration algorithms, these algorithms offer greater stability and accuracy. Several asymptotically correct, thereby enabling retention of stability and accuracy when large increments of independent variable used. Accuracies attainable demonstrated by applying them to systems of nonlinear, first-order, differential equations that arise in study of viscoplastic behavior, spread of acquired immune-deficiency syndrome (AIDS) virus and predator/prey populations.
Ultrashort Pulse Propagation in Nonlinear Dispersive Fibers
NASA Astrophysics Data System (ADS)
Agrawal, Govind P.
Ultrashort optical pulses are often propagated through optical waveguides for a variety of applications including telecommunications and supercontinuum generation [1]. Typically the waveguide is in the form of an optical fiber but it can also be a planar waveguide. The material used to make the waveguide is often silica glass, but other materials such as silicon or chalcogenides have also been used in recent years. What is common to all such materials is they exhibit chromatic dispersion as well as the Kerr nonlinearity. The former makes the refractive index frequency dependent, whereas the latter makes it to depend on the intensity of light propagating through the medium [2]. Both of these effects become more important as optical pulses become shorter and more intense. For pulses not too short (pulse widths > 1 ns) and not too intense (peak powers < 10 mW), the waveguide plays a passive role (except for small optical losses) and acts as a transporter of optical pulses from one place to another, without significantly affecting their shape or spectrum. However, as pulses become shorter and more intense, both the dispersion and the Kerr nonlinearity start to affect the shape and spectrum of an optical pulse during its propagation inside the waveguide. This chapter focuses on silica fibers but similar results are expected for other waveguides made of different materials
Christodoulides, D N; Joseph, R L
1984-06-01
The propagation of nonlinear optical pulses in fibers is discussed, taking into account physical effects arising from nonlinearity, dispersion, and transverse confinement. The wave equation is solved by treating the radial dependence of the field in an exact way. The conditions supporting bright solitary waves are presented and compared with previous results. PMID:19721553
A dispersive wave equation using nonlocal elasticity
NASA Astrophysics Data System (ADS)
Challamel, Noël; Rakotomanana, Lalaonirina; Le Marrec, Loïc
2009-08-01
Nonlocal continuum mechanics allows one to account for the small length scale effect that becomes significant when dealing with micro- or nano-structures. This Note investigates a model of wave propagation in a nonlocal elastic material. We show that a dispersive wave equation is obtained from a nonlocal elastic constitutive law, based on a mixture of a local and a nonlocal strain. This model comprises both the classical gradient model and the Eringen's integral model. The dynamic properties of the model are discussed, and corroborate well some recent theoretical studies published to unify both static and dynamics gradient elasticity theories. Moreover, an excellent matching of the dispersive curve of the Born-Kármán model of lattice dynamics is obtained with such nonlocal model. To cite this article: N. Challamel et al., C. R. Mecanique 337 (2009).
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.
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.
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.
Derivation of an Applied Nonlinear Schroedinger Equation.
Pitts, Todd Alan; Laine, Mark Richard; Schwarz, Jens; Rambo, Patrick K.; Karelitz, David B.
2015-01-01
We derive from first principles a mathematical physics model useful for understanding nonlinear optical propagation (including filamentation). All assumptions necessary for the development are clearly explained. We include the Kerr effect, Raman scattering, and ionization (as well as linear and nonlinear shock, diffraction and dispersion). We explain the phenomenological sub-models and each assumption required to arrive at a complete and consistent theoretical description. The development includes the relationship between shock and ionization and demonstrates why inclusion of Drude model impedance effects alters the nature of the shock operator. Unclassified Unlimited Release
NASA Astrophysics Data System (ADS)
Thomson, Mark J.; McKellar, Bruce H. J.
1991-04-01
A simple, non-linear generalization of the MSW equation is presented and its analytic solution is outlined. The orbits of the polarization vector are shown to be periodic, and to lie on a sphere. Their non-trivial flow patterns fall into two topological categories, the more complex of which can become chaotic if perturbed.
Quantum nonlinear Schrodinger equation on a lattice
Bogolyubov, N.M.; Korepin, V.E.
1986-09-01
A local Hamiltonian is constructed for the nonlinear Schrodinger equation on a lattice in both the classical and the quantum variants. This Hamiltonian is an explicit elementary function of the local Bose fields. The lattice model possesses the same structure of the action-angle variables as the continuous model.
NASA Astrophysics Data System (ADS)
Ntsime, Basetsana P.; Moitsheki, Raseelo J.
2016-06-01
In this paper we consider a nonlinear convection-dispersion equation arising in contaminant transport. The water flow velocity is considered to be spatially-dependent and dispersion coefficient depends on concentration. A direct group classification resulted in a number of cases for which the governing equation admits Lie point symmetries. In each case the one dimensional optimal system of subalgebras is constructed. Reductions are performed. The reduced ordinary differential equations (ODEs) are nonlinear and difficult to solve exactly. On the other hand we consider the steady state problem and applied the method of canonical coordinates to determine exact solutions.
Amplitude-dependent Lamb wave dispersion in nonlinear plates.
Packo, Pawel; Uhl, Tadeusz; Staszewski, Wieslaw J; Leamy, Michael J
2016-08-01
The paper presents a perturbation approach for calculating amplitude-dependent Lamb wave dispersion in nonlinear plates. Nonlinear dispersion relationships are derived in closed form using a hyperelastic stress-strain constitutive relationship, the Green-Lagrange strain measure, and the partial wave technique integrated with a Lindstedt-Poincaré perturbation approach. Solvability conditions are derived using an operator formalism with inner product projections applied against solutions to the adjoint problem. When applied to the first- and second-order problems, these solvability conditions lead to amplitude-dependent, nonlinear dispersion corrections for frequency as a function of wavenumber. Numerical simulations verify the predicted dispersion shifts for an example nonlinear plate. The analysis and identification of amplitude-dependent, nonlinear Lamb wave dispersion complements recent research focusing on higher harmonic generation and internally resonant waves, which require precise dispersion relationships for frequency-wavenumber matching. PMID:27586758
Solitary waves in the nonlinear Dirac equation with arbitrary nonlinearity.
Cooper, Fred; Khare, Avinash; Mihaila, Bogdan; Saxena, Avadh
2010-09-01
We consider the nonlinear Dirac equations (NLDE's) in 1+1 dimension with scalar-scalar self interaction g{2}/k+1(ΨΨ){k+1} , as well as a vector-vector self interaction g{2}/k+1(Ψγ{μ}ΨΨγ{μ}Ψ){1/2(k+1)} . We find the exact analytic form for solitary waves for arbitrary k and find that they are a generalization of the exact solutions for the nonlinear Schrödinger equation (NLSE) and reduce to these solutions in a well defined nonrelativistic limit. We perform the nonrelativistic reduction and find the 1/2m correction to the NLSE, valid when |ω-m|<2m , where ω is the frequency of the solitary wave in the rest frame. We discuss the stability and blowup of solitary waves assuming the modified NLSE is valid and find that they should be stable for k<2 . PMID:21230200
Soliton solutions to a few fractional nonlinear evolution equations in shallow water wave dynamics
NASA Astrophysics Data System (ADS)
Mirzazadeh, Mohammad; Ekici, Mehmet; Sonmezoglu, Abdullah; Ortakaya, Sami; Eslami, Mostafa; Biswas, Anjan
2016-05-01
This paper studies a few nonlinear evolution equations that appear with fractional temporal evolution and fractional spatial derivatives. These are Benjamin-Bona-Mahoney equation, dispersive long wave equation and Nizhnik-Novikov-Veselov equation. The extended Jacobi's elliptic function expansion method is implemented to obtain soliton and other periodic singular solutions to these equations. In the limiting case, when the modulus of ellipticity approaches zero or unity, these doubly periodic functions approach solitary waves or shock waves or periodic singular solutions emerge.
Nonlinear, dispersive, elliptically polarized Alfven wavaes
NASA Technical Reports Server (NTRS)
Kennel, C. F.; Buti, B.; Hada, T.; Pellat, R.
1988-01-01
The derivative nonlinear Schroedinger (DNLS) equation is derived by an efficient means that employs Lagrangian variables. An expression for the stationary wave solutions of the DNLS that contains vanishing and nonvanishing and modulated and nonmodulated boundary conditions as subcases is then obtained. The solitary wave solutions for elliptically polarized quasiparallel Alfven waves in the magnetohydrodynamic limit (nonvanishing, unmodulated boundary conditions) are obtained. These converge to the Korteweg-de Vries and the modified Korteweg-de Vries solitons obtained previously for oblique propagation, but are more general. It is shown that there are no envelope solitary waves if the point at infinity is unstable to the modulational instability. The periodic solutions of the DNLS are characterized.
Taming the nonlinearity of the Einstein equation.
Harte, Abraham I
2014-12-31
Many of the technical complications associated with the general theory of relativity ultimately stem from the nonlinearity of Einstein's equation. It is shown here that an appropriate choice of dynamical variables may be used to eliminate all such nonlinearities beyond a particular order: Both Landau-Lifshitz and tetrad formulations of Einstein's equation are obtained that involve only finite products of the unknowns and their derivatives. Considerable additional simplifications arise in physically interesting cases where metrics become approximately Kerr or, e.g., plane waves, suggesting that the variables described here can be used to efficiently reformulate perturbation theory in a variety of contexts. In all cases, these variables are shown to have simple geometrical interpretations that directly relate the local causal structure associated with the metric of interest to the causal structure associated with a prescribed background. A new method to search for exact solutions is outlined as well. PMID:25615299
Dispersion equation of gravito-MHD waves
NASA Astrophysics Data System (ADS)
Jovanović, Gordana
2016-03-01
We derive the dispersion equation for gravito-MHD waves in an isothermal, gravitationally stratified plasma with a horizontal inhomogeneous magnetic field. In the present model the sound and the Alfvén speeds are constant. It is known that in this model analytical solutions can be obtained for linearized perturbations. There are three modes propagating in the considered plasma: the fast, the slow and the Alfvén mode, all modified by gravity. In the extreme short wavelength limit, these waves propagate in a locally uniform plasma. The waves with larger wavelengths will be affected by the nonuniformity of the medium resulting from the action of gravitational force ρg. In the case without magnetic field these waves become gravito-acoustic waves.
Explicit integration of Friedmann's equation with nonlinear equations of state
NASA Astrophysics Data System (ADS)
Chen, Shouxin; Gibbons, Gary W.; Yang, Yisong
2015-05-01
In this paper we study the integrability of the Friedmann equations, when the equation of state for the perfect-fluid universe is nonlinear, in the light of the Chebyshev theorem. A series of important, yet not previously touched, problems will be worked out which include the generalized Chaplygin gas, two-term energy density, trinomial Friedmann, Born-Infeld, two-fluid models, and Chern-Simons modified gravity theory models. With the explicit integration, we are able to understand exactly the roles of the physical parameters in various models play in the cosmological evolution which may also offer clues to a profound understanding of the problems in general settings. For example, in the Chaplygin gas universe, a few integrable cases lead us to derive a universal formula for the asymptotic exponential growth rate of the scale factor, of an explicit form, whether the Friedmann equation is integrable or not, which reveals the coupled roles played by various physical sectors and it is seen that, as far as there is a tiny presence of nonlinear matter, conventional linear matter makes contribution to the dark matter, which becomes significant near the phantom divide line. The Friedmann equations also arise in areas of physics not directly related to cosmology. We provide some examples ranging from geometric optics and central orbits to soap films and the shape of glaciated valleys to which our results may be applied.
Solving Nonlinear Euler Equations with Arbitrary Accuracy
NASA Technical Reports Server (NTRS)
Dyson, Rodger W.
2005-01-01
A computer program that efficiently solves the time-dependent, nonlinear Euler equations in two dimensions to an arbitrarily high order of accuracy has been developed. The program implements a modified form of a prior arbitrary- accuracy simulation algorithm that is a member of the class of algorithms known in the art as modified expansion solution approximation (MESA) schemes. Whereas millions of lines of code were needed to implement the prior MESA algorithm, it is possible to implement the present MESA algorithm by use of one or a few pages of Fortran code, the exact amount depending on the specific application. The ability to solve the Euler equations to arbitrarily high accuracy is especially beneficial in simulations of aeroacoustic effects in settings in which fully nonlinear behavior is expected - for example, at stagnation points of fan blades, where linearizing assumptions break down. At these locations, it is necessary to solve the full nonlinear Euler equations, and inasmuch as the acoustical energy is of the order of 4 to 5 orders of magnitude below that of the mean flow, it is necessary to achieve an overall fractional error of less than 10-6 in order to faithfully simulate entropy, vortical, and acoustical waves.
Exact kink solitons in the presence of diffusion, dispersion, and polynomial nonlinearity
NASA Astrophysics Data System (ADS)
Raposo, E. P.; Bazeia, D.
1999-03-01
We describe exact travelling-wave kink soliton solutions in some classes of nonlinear partial differential equations, such as generalized Korteweg-de Vries-Burgers, Korteweg-de Vries-Huxley, and Korteweg-de Vries-Burgers-Huxley equations, as well as equations in the generic form ut + P( u) ux + vuxx - δuxxx = A( u), with polynomial functions P( u) and A( u) of u = u( x, t), whose generality allows the identification with a number of relevant equations in physics. We focus on the analysis of the role of diffusion, dispersion, nonlinear effects, and parity of the polynomials to the properties of the solutions, particularly their velocity of propagation. In addition, we show that, for some appropriate choices, these equations can be mapped onto equations of motion of relativistic (1 + 1)-dimensional φ4 and φ6 field theories of real scalar fields. Systems of two coupled nonlinear equations are also considered.
Dispersion and nonlinear effects in the 2011 Tohoku-Oki earthquake tsunami
NASA Astrophysics Data System (ADS)
Saito, Tatsuhiko; Inazu, Daisuke; Miyoshi, Takayuki; Hino, Ryota
2014-08-01
This study reveals the roles of the wave dispersion and nonlinear effects for the 2011 Tohoku-Oki earthquake tsunami. We conducted tsunami simulations based on the nonlinear dispersive equations with a high-resolution source model. The simulations successfully reproduced the waveforms recorded in the offshore, deep sea, and focal areas. The calculated inundation area coincided well with the actual inundation for the Sendai Plain, which was the widest inundation area during this event. By conducting sets of simulations with different tsunami equations, we obtained the followings insights into the wave dispersion, nonlinear effects, and energy dissipation for this event. Although the wave dispersion was neglected in most studies, the maximum amplitude was significantly overestimated in the deep sea if the dispersion was not included. The waveform observed at the station with the largest tsunami height (˜2 m) among the deep-ocean stations also verified the necessity of the dispersion. It is well known that the nonlinear effects play an important role for the propagation of a tsunami into bays and harbors. Additionally, nonlinear effects need to be considered to accurately model later waves, even for offshore stations. In particular, including nonlinear terms rather than the inundation was more important when precisely modeling the waves reflected from the coast.
Forces Associated with Nonlinear Nonholonomic Constraint Equations
NASA Technical Reports Server (NTRS)
Roithmayr, Carlos M.; Hodges, Dewey H.
2010-01-01
A concise method has been formulated for identifying a set of forces needed to constrain the behavior of a mechanical system, modeled as a set of particles and rigid bodies, when it is subject to motion constraints described by nonholonomic equations that are inherently nonlinear in velocity. An expression in vector form is obtained for each force; a direction is determined, together with the point of application. This result is a consequence of expressing constraint equations in terms of dot products of vectors rather than in the usual way, which is entirely in terms of scalars and matrices. The constraint forces in vector form are used together with two new analytical approaches for deriving equations governing motion of a system subject to such constraints. If constraint forces are of interest they can be brought into evidence in explicit dynamical equations by employing the well-known nonholonomic partial velocities associated with Kane's method; if they are not of interest, equations can be formed instead with the aid of vectors introduced here as nonholonomic partial accelerations. When the analyst requires only the latter, smaller set of equations, they can be formed directly; it is not necessary to expend the labor to form the former, larger set first and subsequently perform matrix multiplications.
Dark soliton solutions of (N+1)-dimensional nonlinear evolution equations
NASA Astrophysics Data System (ADS)
Demiray, Seyma Tuluce; Bulut, Hasan
2016-06-01
In this study, we investigate exact solutions of (N+1)-dimensional double sinh-Gordon equation and (N+1)-dimensional generalized Boussinesq equation by using generalized Kudryashov method. (N+1)-dimensional double sinh-Gordon equation and (N+1)-dimensional generalized Boussinesq equation can be returned to nonlinear ordinary differential equation by suitable transformation. Then, generalized Kudryashov method has been used to seek exact solutions of the (N+1)-dimensional double sinh-Gordon equation and (N+1)-dimensional generalized Boussinesq equation. Also, we obtain dark soliton solutions for these (N+1)-dimensional nonlinear evolution equations. Finally, we denote that this method can be applied to solve other nonlinear evolution equations.
Using the scalable nonlinear equations solvers package
Gropp, W.D.; McInnes, L.C.; Smith, B.F.
1995-02-01
SNES (Scalable Nonlinear Equations Solvers) is a software package for the numerical solution of large-scale systems of nonlinear equations on both uniprocessors and parallel architectures. SNES also contains a component for the solution of unconstrained minimization problems, called SUMS (Scalable Unconstrained Minimization Solvers). Newton-like methods, which are known for their efficiency and robustness, constitute the core of the package. As part of the multilevel PETSc library, SNES incorporates many features and options from other parts of PETSc. In keeping with the spirit of the PETSc library, the nonlinear solution routines are data-structure-neutral, making them flexible and easily extensible. This users guide contains a detailed description of uniprocessor usage of SNES, with some added comments regarding multiprocessor usage. At this time the parallel version is undergoing refinement and extension, as we work toward a common interface for the uniprocessor and parallel cases. Thus, forthcoming versions of the software will contain additional features, and changes to parallel interface may result at any time. The new parallel version will employ the MPI (Message Passing Interface) standard for interprocessor communication. Since most of these details will be hidden, users will need to perform only minimal message-passing programming.
Nonlinear coupling of left and right handed circularly polarized dispersive Alfvén wave
Sharma, R. P. Sharma, Swati Gaur, Nidhi
2014-07-15
The nonlinear phenomena are of prominent interests in understanding the particle acceleration and transportation in the interplanetary space. The ponderomotive nonlinearity causing the filamentation of the parallel propagating circularly polarized dispersive Alfvén wave having a finite frequency may be one of the mechanisms that contribute to the heating of the plasmas. The contribution will be different of the left (L) handed mode, the right (R) handed mode, and the mix mode. The contribution also depends upon the finite frequency of the circularly polarized waves. In the present paper, we have investigated the effect of the nonlinear coupling of the L and R circularly polarized dispersive Alfvén wave on the localized structures formation and the respective power spectra. The dynamical equations are derived in the presence of the ponderomotive nonlinearity of the L and R pumps and then studied semi-analytically as well as numerically. The ponderomotive nonlinearity accounts for the nonlinear coupling between both the modes. In the presence of the adiabatic response of the density fluctuations, the nonlinear dynamical equations satisfy the modified nonlinear Schrödinger equation. The equations thus obtained are solved in solar wind regime to study the coupling effect on localization and the power spectra. The effect of coupling is also studied on Faraday rotation and ellipticity of the wave caused due to the difference in the localization of the left and the right modes with the distance of propagation.
The beam equation with nonlinear memory
NASA Astrophysics Data System (ADS)
D'Abbicco, Marcello; Lucente, Sandra
2016-06-01
In this paper, we study the critical exponent for the beam equation with nonlinear memory, i.e., {u_{tt}+Δ^2u = F(t, u)}, where F = intlimits0tf(t - s)N(u)(s, x) {d}s, quad N(u)≈ |u|^p. For suitable f and p, we prove the existence of local-in-time solutions and small data global solutions to the Cauchy problem, in homogeneous and nonhomogeneous Sobolev spaces. In some cases, we prove that the local solution cannot be extended to a global one. We also consider the limit case of power nonlinearity, i.e., {F = N(u)}.
Dispersion of Sound in Dilute Suspensions with Nonlinear Particle Relaxation
NASA Technical Reports Server (NTRS)
Kandula, Max
2010-01-01
The theory accounting for nonlinear particle relaxation (viscous and thermal) has been applied to the prediction of dispersion of sound in dilute suspensions. The results suggest that significant deviations exist for sound dispersion between the linear and nonlinear theories at large values of Omega(Tau)(sub d), where Omega is the circular frequency, and Tau(sub d) is the Stokesian particle relaxation time. It is revealed that the nonlinear effect on the dispersion coefficient due to viscous contribution is larger relative to that of thermal conduction
Stochastic differential equations and turbulent dispersion
NASA Technical Reports Server (NTRS)
Durbin, P. A.
1983-01-01
Aspects of the theory of continuous stochastic processes that seem to contribute to an understanding of turbulent dispersion are introduced and the theory and philosophy of modelling turbulent transport is emphasized. Examples of eddy diffusion examined include shear dispersion, the surface layer, and channel flow. Modeling dispersion with finite-time scale is considered including the Langevin model for homogeneous turbulence, dispersion in nonhomogeneous turbulence, and the asymptotic behavior of the Langevin model for nonhomogeneous turbulence.
Exact and explicit solitary wave solutions to some nonlinear equations
Jiefang Zhang
1996-08-01
Exact and explicit solitary wave solutions are obtained for some physically interesting nonlinear evolutions and wave equations in physics and other fields by using a special transformation. These equations include the KdV-Burgers equation, the MKdV-Burgers equation, the combined KdV-MKdV equation, the Newell-Whitehead equation, the dissipative {Phi}{sup 4}-model equation, the generalized Fisher equation, and the elastic-medium wave equation.
FAST TRACK COMMUNICATION Quasi self-adjoint nonlinear wave equations
NASA Astrophysics Data System (ADS)
Ibragimov, N. H.; Torrisi, M.; Tracinà, R.
2010-11-01
In this paper we generalize the classification of self-adjoint second-order linear partial differential equation to a family of nonlinear wave equations with two independent variables. We find a class of quasi self-adjoint nonlinear equations which includes the self-adjoint linear equations as a particular case. The property of a differential equation to be quasi self-adjoint is important, e.g. for constructing conservation laws associated with symmetries of the differential equation.
Solution spectrum of nonlinear diffusion equations
Ulmer, W.
1992-08-01
The stationary version of the nonlinear diffusion equation -{partial_derivative}c/{partial_derivative}t+D{Delta}c=A{sub 1}c-A{sub 2}c{sup 2} can be solved with the ansatz c={summation}{sub p=1}{sup {infinity}} A{sub p}(cosh kx){sup -p}, inducing a band structure with regard to the ratio {lambda}{sub 1}/{lambda}{sub 2}. The resulting solution manifold can be related to an equilibrium of fluxes of nonequilibrium thermodynamics. The modification of this ansatz yielding the expansion c={summation}{sub p,q=1}{sup infinity}A{sub pa}(cosh kx){sup -p}[(cosh {alpha}t){sup -q-1} sinh {alpha}t+b(cosh {alpha}t){sup -q}] represents a solution spectrum of the time-dependent nonlinear equations, and the stationary version can be found from the asymptotic behaviour of the expansion. The solutions can be associated with reactive processes such as active transport phenomena and control circuit problems is discussed. There are also applications to cellular kinetics of clonogenic cell assays and spheriods. 33 refs., 1 tab.
Stationary states of extended nonlinear Schrödinger equation with a source
NASA Astrophysics Data System (ADS)
Borich, M. A.; Smagin, V. V.; Tankeev, A. P.
2007-02-01
Structure of nonlinear stationary states of the extended nonlinear Schrödinger equation (ENSE) with a source has been analyzed with allowance for both third-order and nonlinearity dispersion. A new class of particular solutions (solitary waves) of the ENSe has been obtained. The scenario of the destruction of these states under the effect of an external perturbation has been investigated analytically and numerically. The results obtained can be used to interpret experimental data on the weakly nonlinear dynamics of the magnetostatic envelope in heterophase ferromagnet-insulator-metal, metal-insulator-ferromagnet-insulator-metal, and other similar structures and upon the simulation of nonlinear processes in optical systems.
Nonlinear scattering term in the gyrokinetic Vlasov equation
Wang, Shaojie
2013-08-15
Nonlinear scattering term is found from the nonlinear gyrokinetic equation by decoupling the perturbed gyrocenter motion from the unperturbed motion. The gyro-center distribution function is determined by the well-understood unperturbed motion, with the effects of fields perturbation included in the nonlinear scattering term, which explicitly reveals the nonlinear stochastic dissipation on the time scale longer than the wave correlation time.
Soliton dispersion management in nonlinear optical fibers
NASA Astrophysics Data System (ADS)
Ganapathy, R.
2012-12-01
We consider the concept of quasisoliton propagation in a dispersion management fiber and study the soliton dynamics for soliton dispersion management case, soliton energy control case and guiding center soliton case. We also study the interaction scenario in detail for all the cases.
Power-law spatial dispersion from fractional Liouville equation
Tarasov, Vasily E.
2013-10-15
A microscopic model in the framework of fractional kinetics to describe spatial dispersion of power-law type is suggested. The Liouville equation with the Caputo fractional derivatives is used to obtain the power-law dependence of the absolute permittivity on the wave vector. The fractional differential equations for electrostatic potential in the media with power-law spatial dispersion are derived. The particular solutions of these equations for the electric potential of point charge in this media are considered.
Numerical solutions of Maxwell's equations for nonlinear-optical pulse propagation
NASA Astrophysics Data System (ADS)
Hile, Cheryl V.; Kath, William L.
1996-06-01
A model and numerical solutions of Maxwell's equations describing the propagation of short, solitonlike pulses in nonlinear dispersive optical media are presented. The model includes linear dispersion expressed in the time domain, a Kerr nonlinearity, and a coordinate system moving with the group velocity of the pulse. Numerical solutions of Maxwell's equations are presented for circularly polarized and linearly polarized electromagnetic fields. When the electromagnetic fields are assumed to be circularly polarized, numerical solutions are compared directly with solutions of the nonlinear Schrodinger (NLS) equation. These comparisons show good agreement and indicate that the NLS equation provides an excellent model for short-pulse propagation. When the electromagnetic fields are assumed to be linearly polarized, the propagation of daughter pulses, small-amplitude pulses at three times the frequency of the solitonlike pulse, are observed in the numerical solution. These daughter pulses are shown to be the direct result of third harmonics generated by the main, solitonlike, pulse.
Controlling Spatiotemporal Chaos in Active Dissipative-Dispersive Nonlinear Systems
NASA Astrophysics Data System (ADS)
Gomes, Susana; Pradas, Marc; Kalliadasis, Serafim; Papageorgiou, Demetrios; Pavliotis, Grigorios
2015-11-01
We present a novel generic methodology for the stabilization and control of infinite-dimensional dynamical systems exhibiting low-dimensional spatiotemporal chaos. The methodology is exemplified with the generalized Kuramoto-Sivashinsky equation, the simplest possible prototype that retains that fundamental elements of any nonlinear process involving wave evolution. The equation is applicable on a wide variety of systems including falling liquid films and plasma waves with dispersion due to finite banana width. We show that applying the appropriate choice of time-dependent feedback controls via blowing and suction, we are able to stabilize and/or control all stable or unstable solutions, including steady solutions, travelling waves and spatiotemporal chaos, but also use the controls obtained to stabilize the solutions to more general long wave models. We acknowledge financial support from Imperial College through a Roth PhD studentship, Engineering and Physical Sciences Research Council of the UK through Grants No. EP/H034587, EP/J009636, EP/K041134, EP/L020564 and EP/L024926 and European Research Council via Advanced Grant No. 247031.
Complex PT-symmetric nonlinear Schrödinger equation and Burgers equation.
Yan, Zhenya
2013-04-28
The complex -symmetric nonlinear wave models have drawn much attention in recent years since the complex -symmetric extensions of the Korteweg-de Vries (KdV) equation were presented in 2007. In this review, we focus on the study of the complex -symmetric nonlinear Schrödinger equation and Burgers equation. First of all, we briefly introduce the basic property of complex symmetry. We then report on exact solutions of one- and two-dimensional nonlinear Schrödinger equations (known as the Gross-Pitaevskii equation in Bose-Einstein condensates) with several complex -symmetric potentials. Finally, some complex -symmetric extension principles are used to generate some complex -symmetric nonlinear wave equations starting from both -symmetric (e.g. the KdV equation) and non- -symmetric (e.g. the Burgers equation) nonlinear wave equations. In particular, we discuss exact solutions of some representative ones of the complex -symmetric Burgers equation in detail. PMID:23509385
Realistic dispersion kernels applied to cohabitation reaction dispersion equations
NASA Astrophysics Data System (ADS)
Isern, Neus; Fort, Joaquim; Pérez-Losada, Joaquim
2008-10-01
We develop front spreading models for several jump distance probability distributions (dispersion kernels). We derive expressions for a cohabitation model (cohabitation of parents and children) and a non-cohabitation model, and apply them to the Neolithic using data from real human populations. The speeds that we obtain are consistent with observations of the Neolithic transition. The correction due to the cohabitation effect is up to 38%.
On shallow water waves in a medium with time-dependent dispersion and nonlinearity coefficients
Abdel-Gawad, Hamdy I.; Osman, Mohamed
2014-01-01
In this paper, we studied the progression of shallow water waves relevant to the variable coefficient Korteweg–de Vries (vcKdV) equation. We investigated two kinds of cases: when the dispersion and nonlinearity coefficients are proportional, and when they are not linearly dependent. In the first case, it was shown that the progressive waves have some geometric structures as in the case of KdV equation with constant coefficients but the waves travel with time dependent speed. In the second case, the wave structure is maintained when the nonlinearity balances the dispersion. Otherwise, water waves collapse. The objectives of the study are to find a wide class of exact solutions by using the extended unified method and to present a new algorithm for treating the coupled nonlinear PDE’s. PMID:26199750
On shallow water waves in a medium with time-dependent dispersion and nonlinearity coefficients.
Abdel-Gawad, Hamdy I; Osman, Mohamed
2015-07-01
In this paper, we studied the progression of shallow water waves relevant to the variable coefficient Korteweg-de Vries (vcKdV) equation. We investigated two kinds of cases: when the dispersion and nonlinearity coefficients are proportional, and when they are not linearly dependent. In the first case, it was shown that the progressive waves have some geometric structures as in the case of KdV equation with constant coefficients but the waves travel with time dependent speed. In the second case, the wave structure is maintained when the nonlinearity balances the dispersion. Otherwise, water waves collapse. The objectives of the study are to find a wide class of exact solutions by using the extended unified method and to present a new algorithm for treating the coupled nonlinear PDE's. PMID:26199750
Forced nonlinear Schrödinger equation with arbitrary nonlinearity
NASA Astrophysics Data System (ADS)
Cooper, Fred; Khare, Avinash; Quintero, Niurka R.; Mertens, Franz G.; Saxena, Avadh
2012-04-01
We consider the nonlinear Schrödinger equation (NLSE) in 1+1 dimension with scalar-scalar self-interaction (g2)/(κ+1)(ψψ)κ+1 in the presence of the external forcing terms of the form re-i(kx+θ)-δψ. We find new exact solutions for this problem and show that the solitary wave momentum is conserved in a moving frame where vk=2k. These new exact solutions reduce to the constant phase solutions of the unforced problem when r→0. In particular we study the behavior of solitary wave solutions in the presence of these external forces in a variational approximation which allows the position, momentum, width, and phase of these waves to vary in time. We show that the stationary solutions of the variational equations include a solution close to the exact one and we study small oscillations around all the stationary solutions. We postulate that the dynamical condition for instability is that dp(t)/dq˙(t)<0, where p(t) is the normalized canonical momentum p(t)=(1)/(M(t))(∂L)/(∂q˙), and q˙(t) is the solitary wave velocity. Here M(t)=∫dxψ(x,t)ψ(x,t). Stability is also studied using a “phase portrait” of the soliton, where its dynamics is represented by two-dimensional projections of its trajectory in the four-dimensional space of collective coordinates. The criterion for stability of a soliton is that its trajectory is a closed single curve with a positive sense of rotation around a fixed point. We investigate the accuracy of our variational approximation and these criteria using numerical simulations of the NLSE. We find that our criteria work quite well when the magnitude of the forcing term is small compared to the amplitude of the unforced solitary wave. In this regime the variational approximation captures quite well the behavior of the solitary wave.
Low-Dispersion Scheme for Nonlinear Acoustic Waves in Nonuniform Flow
NASA Technical Reports Server (NTRS)
Baysal, Oktay; Kaushik, Dinesh K.; Idres, Moumen
1997-01-01
The linear dispersion-relation-preserving scheme and its boundary conditions have been extended to the nonlinear Euler equations. This allowed computing, a nonuniform flowfield and a nonlinear acoustic wave propagation in such a medium, by the same scheme. By casting all the equations, boundary conditions, and the solution scheme in generalized curvilinear coordinates, the solutions were made possible for non-Cartesian domains and, for the better deployment of the grid points, nonuniform grid step sizes could be used. It has been tested for a number of simple initial-value and periodic-source problems. A simple demonstration of the difference between a linear and nonlinear propagation was conducted. The wall boundary condition, derived from the momentum equations and implemented through a pressure at a ghost point, and the radiation boundary condition, derived from the asymptotic solution to the Euler equations, have proven to be effective for the nonlinear equations and nonuniform flows. The nonreflective characteristic boundary conditions also have shown success but limited to the nonlinear waves in no mean flow, and failed for nonlinear waves in nonuniform flow.
Wave-packet formation at the zero-dispersion point in the Gardner-Ostrovsky equation.
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. PMID:26066112
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
Crosta, M.; Fratalocchi, A.; Trillo, S.
2011-12-15
We characterize the full family of soliton solutions sitting over a background plane wave and ruled by the cubic-quintic nonlinear Schroedinger equation in the regime where a quintic focusing term represents a saturation of the cubic defocusing nonlinearity. We discuss the existence and properties of solitons in terms of catastrophe theory and fully characterize bistability and instabilities of the dark-antidark pairs, revealing mechanisms of decay of antidark solitons into dispersive shock waves.
Capillary waves in the subcritical nonlinear Schroedinger equation
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.
Perturbation approach to dispersion curves calculation for nonlinear Lamb waves
NASA Astrophysics Data System (ADS)
Packo, Pawel; Staszewski, Wieslaw J.; Uhl, Tadeusz; Leamy, Michael J.
2015-05-01
Analysis of elastic wave propagation in nonlinear media has gained recent research attention due to the recognition of their amplitude-dependent behavior. This creates opportunities for increased accuracy of damage detection and localization, development of new structural monitoring strategies, and design of new structures with desirable acoustic behavior (e.g., amplitude-dependent frequency bandgaps, wave beaming, and filtering). This differs from more traditional nonlinear analysis approaches which target the prediction of higher harmonic growth. Of particular interest in this work is the analysis of amplitude-dependent shifts in Lamb wave dispersion curves. Typically, dispersion curves are calculated for nominally linear material parameters and geometrical features of a waveguide, even when the constitutive law is nonlinear. Instead, this work employs a Lindstedt - Poincare perturbation approach to calculate amplitude-dependent dispersion curves, and shifts thereof, for nonlinearly-elastic plates. As a result, a set of first order corrections to frequency (or equivalently wavenumber) are calculated. These corrections yield significant amplitude dependence in the spectral characteristics of the calculated waves, especially for high frequency waves, which differs fundamentally from linear analyses. Numerical simulations confirm the analytical shifts predicted. Recognition of this amplitude-dependence in Lamb wave dispersion may suggest, among other things, that the analysis of guided wave propagation phenomena within a fully nonlinear framework needs to revisit mode-mode energy flux and higher harmonics generation conditions.
On the nonlinear Schrodinger equation with nonzero boundary conditions
NASA Astrophysics Data System (ADS)
Fagerstrom, Emily
This thesis is concerned with the study of the nonlinear Schrodinger (NLS) equation, which is important both from a physical and a mathematical point of view. In physics, it is a universal model for the evolutions of weakly nonlinear dispersive wave trains. As such it appears in many physical contexts, such as optics, acoustics, plasmas, biology, etc. Mathematically, it is a completely integrable, infinite-dimensional Hamiltonian system, and possesses a surprisingly rich structure. This equation has been extensively studied in the last 50 years, but many important questions are still open. In particular, this thesis contains the following original contributions: NLS with real spectral singularities. First, the focusing NLS equation is considered with decaying initial conditions. This situation has been studied extensively before, but the assumption is almost always made that the scattering coefficients have no real zeros, and thus the scattering data had no poles on the real axis. However, it is easy to produce example potentials with this behavior. For example, by modifying parameters in Satsuma-Yajima's sech potential, or by choosing a "box" potential with a particular area, one can obtain corresponding scattering entries with real zeros. The inverse scattering transform can be implemented by formulating the modified Jost eigenfunctions and the scattering data as a Riemann Hilbert problem. But it can also be formulated by using integral kernels. Doing so produces the Gelf'and-Levitan-Marchenko (GLM) equations. Solving these integral equations requires integrating an expression containing the reflection coefficient over the real axis. Under the usual assumption, the reflection coefficient has no poles on the real axis. In general, the integration contour cannot be deformed to avoid poles, because the reflection coefficient may not admit analytic extension off the real axis. Here it is shown that the GLM equations may be (uniquely) solved using a principal value
Forced nonlinear Schrödinger equation with arbitrary nonlinearity.
Cooper, Fred; Khare, Avinash; Quintero, Niurka R; Mertens, Franz G; Saxena, Avadh
2012-04-01
We consider the nonlinear Schrödinger equation (NLSE) in 1+1 dimension with scalar-scalar self-interaction g(2)/κ+1(ψ*ψ)(κ+1) in the presence of the external forcing terms of the form re(-i(kx+θ))-δψ. We find new exact solutions for this problem and show that the solitary wave momentum is conserved in a moving frame where v(k)=2k. These new exact solutions reduce to the constant phase solutions of the unforced problem when r→0. In particular we study the behavior of solitary wave solutions in the presence of these external forces in a variational approximation which allows the position, momentum, width, and phase of these waves to vary in time. We show that the stationary solutions of the variational equations include a solution close to the exact one and we study small oscillations around all the stationary solutions. We postulate that the dynamical condition for instability is that dp(t)/dq ̇(t)<0, where p(t) is the normalized canonical momentum p(t)=1/M(t)∂L/∂q ̇, and q ̇(t) is the solitary wave velocity. Here M(t)=∫dxψ*(x,t)ψ(x,t). Stability is also studied using a "phase portrait" of the soliton, where its dynamics is represented by two-dimensional projections of its trajectory in the four-dimensional space of collective coordinates. The criterion for stability of a soliton is that its trajectory is a closed single curve with a positive sense of rotation around a fixed point. We investigate the accuracy of our variational approximation and these criteria using numerical simulations of the NLSE. We find that our criteria work quite well when the magnitude of the forcing term is small compared to the amplitude of the unforced solitary wave. In this regime the variational approximation captures quite well the behavior of the solitary wave. PMID:22680598
Exact Travelling Wave Solutions of the Nonlinear Evolution Equations by Auxiliary Equation Method
NASA Astrophysics Data System (ADS)
Kaplan, Melike; Akbulut, Arzu; Bekir, Ahmet
2015-10-01
The auxiliary equation method presents wide applicability to handling nonlinear wave equations. In this article, we establish new exact travelling wave solutions of the nonlinear Zoomeron equation, coupled Higgs equation, and equal width wave equation. The travelling wave solutions are expressed by the hyperbolic functions, trigonometric functions, and rational functions. It is shown that the proposed method provides a powerful mathematical tool for solving nonlinear wave equations in mathematical physics and engineering. Throughout the article, all calculations are made with the aid of the Maple packet program.
Exact traveling wave solutions for system of nonlinear evolution equations.
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. PMID:27347461
Nonlinear tunneling of optical soliton in 3 coupled NLS equation with symbolic computation
NASA Astrophysics Data System (ADS)
Mani Rajan, M. S.; Mahalingam, A.; Uthayakumar, A.
2014-07-01
We investigated the soliton solution for N coupled nonlinear Schrödinger (CNLS) equations. These equations are coupled due to the cross-phase-modulation (CPM). Lax pair of this system is obtained via the Ablowitz-Kaup-Newell-Segur (AKNS) scheme and the corresponding Darboux transformation is constructed to derive the soliton solution. One and two soliton solutions are generated. Using two soliton solutions of 3 CNLS equation, nonlinear tunneling of soliton for both with and without exponential background has been discussed. Finally cascade compression of optical soliton through multi-nonlinear barrier has been discussed. The obtained results may have promising applications in all-optical devices based on optical solitons, study of soliton propagation in birefringence fiber systems and optical soliton with distributed dispersion and nonlinearity management.
Stochastic differential equations for non-linear hydrodynamics
NASA Astrophysics Data System (ADS)
Español, Pep
1998-02-01
We formulate the stochastic differential equations for non-linear hydrodynamic fluctuations. The equations incorporate the random forces through a random stres tensor and random heat flux as in the Landau and Lifshitz theory. However, the equations are non-linear and the random forces are non-Gaussian. We provide explicit expressions for these random quantities in terms of the well-defined increments of the Wienner process.
Latchio Tiofack, Camus G; Mohamadou, Alidou; Kofané, Timoléon C; Moubissi, Alain B
2009-12-01
We consider a higher-order complex Ginzburg-Landau equation, with the fourth-order dispersion and cubic-quintic nonlinear terms, which can describe the propagation of an ultrashort subpicosecond or femtosecond optical pulse in an optical fiber system. We investigate the modulational instability (MI) of continuous wave solution of this equation. Several types of modulational instability gains are shown to exist in both the anomalous and normal dispersion regimes. We find that depending on the sign of the fourth-order dispersion coefficient, the MI appears for normal and anomalous dispersion regime. Simulations of the full system demonstrate that the development of the MI leads to establishment of a regular or chaotic array of pulses, a chain of well-separated peaks with continuously growing or decaying amplitudes depending on the sign of the loss/gain coefficient and higher-order dispersions terms. Comparison of the calculations with reported numerical results shows a satisfactory agreement. PMID:20365291
Collocation Method for Numerical Solution of Coupled Nonlinear Schroedinger Equation
Ismail, M. S.
2010-09-30
The coupled nonlinear Schroedinger equation models several interesting physical phenomena presents a model equation for optical fiber with linear birefringence. In this paper we use collocation method to solve this equation, we test this method for stability and accuracy. Numerical tests using single soliton and interaction of three solitons are used to test the resulting scheme.
Analytic solutions of a general nonlinear functional equations near resonance
NASA Astrophysics Data System (ADS)
Xu, Bing; Zhang, Weinian
2006-05-01
Existence of analytic solutions of a general class of nonlinear functional equations is discussed. This general class includes some specific functional equations studied recently. Moreover, we can generalize this problem to finding analytic solutions of a general class of iterative equations.
Adiabatic nonlinear waves with trapped particles. II. Wave dispersion
Dodin, I. Y.; Fisch, N. J.
2012-01-15
A general nonlinear dispersion relation is derived in a nondifferential form for an adiabatic sinusoidal Langmuir wave in collisionless plasma, allowing for an arbitrary distribution of trapped electrons. The linear dielectric function is generalized, and the nonlinear kinetic frequency shift {omega}{sub NL} is found analytically as a function of the wave amplitude a. Smooth distributions yield {omega}{sub NL}{proportional_to}{radical}(a), as usual. However, beam-like distributions of trapped electrons result in different power laws, or even a logarithmic nonlinearity, which are derived as asymptotic limits of the same dispersion relation. Such beams are formed whenever the phase velocity changes, because the trapped distribution is in autoresonance and thus evolves differently from the passing distribution. Hence, even adiabatic {omega}{sub NL}(a) is generally nonlocal.
Zhang, Yu-Juan; Zhao, Dun; Luo, Hong-Gang
2014-11-15
We consider a wide class of integrable nonautonomous nonlinear integro-differential Schrödinger equation which contains the models for the soliton management in Bose–Einstein condensates, nonlinear optics, and inhomogeneous Heisenberg spin chain. With the help of the nonisospectral AKNS hierarchy, we obtain the N-fold Darboux transformation and the N-fold soliton-like solutions for the equation. The soliton management, especially the synchronized dispersive and nonlinear management in optical fibers is discussed. It is found that in the situation without external potential, the synchronized dispersive and nonlinear management can keep the integrability of the nonlinear Schrödinger equation; this suggests that in optical fibers, the synchronized dispersive and nonlinear management can control and maintain the propagation of a multi-soliton. - Highlights: • We consider a unified model for soliton management by an integrable integro-differential Schrödinger equation. • Using Lax pair, the N-fold Darboux transformation for the equation is presented. • The multi-soliton management is considered. • The synchronized dispersive and nonlinear management is suggested.
Drift-free kinetic equations for turbulent dispersion.
Bragg, A; Swailes, D C; Skartlien, R
2012-11-01
The dispersion of passive scalars and inertial particles in a turbulent flow can be described in terms of probability density functions (PDFs) defining the statistical distribution of relevant scalar or particle variables. The construction of transport equations governing the evolution of such PDFs has been the subject of numerous studies, and various authors have presented formulations for this type of equation, usually referred to as a kinetic equation. In the literature it is often stated, and widely assumed, that these PDF kinetic equation formulations are equivalent. In this paper it is shown that this is not the case, and the significance of differences among the various forms is considered. In particular, consideration is given to which form of equation is most appropriate for modeling dispersion in inhomogeneous turbulence and most consistent with the underlying particle equation of motion. In this regard the PDF equations for inertial particles are considered in the limit of zero particle Stokes number and assessed against the fully mixed (zero-drift) condition for fluid points. A long-standing question regarding the validity of kinetic equations in the fluid-point limit is answered; it is demonstrated formally that one version of the kinetic equation (derived using the Furutsu-Novikov method) provides a model that satisfies this zero-drift condition exactly in both homogeneous and inhomogeneous systems. In contrast, other forms of the kinetic equation do not satisfy this limit or apply only in a limited regime. PMID:23214875
NA Nonlinear Equation-of-state Inversion
NASA Astrophysics Data System (ADS)
Jackson, I.; Kennett, B. L.
2008-12-01
A fully non-linear inversion scheme is introduced for the determination of the parameters controlling the equation-of-state and elasticity of mineral phases using the thermodynamically consistent finite-strain formulation introduced by Stixrude & Lithgow-Bertelloni (2005). This inversion exploits a directed search in an eight-dimensional parameter space using the Neighbourhood Algorithm (NA) of Sambridge (1999) to search for the minimum of an objective function representing the misfit to multiple data sets that constrain different aspects of the mineral behaviour. No derivatives are employed and the progress towards the minimum builds on the accumulated information on the character of the parameter space acquired as the inversion progresses. When only a limited range of experimental information is available there is a strong possibility of multiple minima in the objective function, which can pose problems for conventional iterative least-squares or other gradient methods. The addition of many different styles of data tends to produce a better defined minimum. The influence of different data types can be readily assessed by allowing differential weighting. The new procedure is illustrated by application to MgO, for which extensive experimental data are available. These include the variation of relative volume V with temperature T and pressure P from both static and shock-compression experiments, acoustic measurements of compressional and shear (and hence bulk) moduli, and calorimetric determinations of entropy as a function of temperature at atmospheric pressure. Preliminary NA modeling highlighted tensions between marginally incompatible subsets of data. We therefore excluded one-atmosphere V(T) data for T ≥ 1800 K for which the quasi-harmonic approximation is inadequate (Wu et al., 2008) along with elastic moduli derived from Brillouin spectroscopy under conditions (P ≥ 14 GPa) where significant departures from hydrostatic conditions are expected. With these
Painlevé equations--nonlinear special functions
NASA Astrophysics Data System (ADS)
Clarkson, Peter A.
2003-04-01
The six Painlevé equations (PI-PVI) were first discovered about a hundred years ago by Painlevé and his colleagues in an investigation of nonlinear second-order ordinary differential equations. Recently, there has been considerable interest in the Painlevé equations primarily due to the fact that they arise as reductions of the soliton equations which are solvable by inverse scattering. Consequently, the Painlevé equations can be regarded as completely integrable equations and possess solutions which can be expressed in terms of solutions of linear integral equations, despite being nonlinear equations. Although first discovered from strictly mathematical considerations, the Painlevé equations have arisen in a variety of important physical applications including statistical mechanics, plasma physics, nonlinear waves, quantum gravity, quantum field theory, general relativity, nonlinear optics and fibre optics. The Painlevé equations may be thought of a nonlinear analogues of the classical special functions. They possess hierarchies of rational solutions and one-parameter families of solutions expressible in terms of the classical special functions, for special values of the parameters. Further the Painlevé equations admit symmetries under affine Weyl groups which are related to the associated Bäcklund transformations. In this paper, I discuss some of the remarkable properties which the Painlevé equations possess including connection formulae, Bäcklund transformations associated discrete equations, and hierarchies of exact solutions. In particular, the second Painlevé equation PII is used to illustrate these properties and some of the applications of PII are also discussed.
Nonlinear tunneling of optical soliton in 3 coupled NLS equation with symbolic computation
Mani Rajan, M.S.; Mahalingam, A.; Uthayakumar, A.
2014-07-15
We investigated the soliton solution for N coupled nonlinear Schrödinger (CNLS) equations. These equations are coupled due to the cross-phase-modulation (CPM). Lax pair of this system is obtained via the Ablowitz–Kaup–Newell–Segur (AKNS) scheme and the corresponding Darboux transformation is constructed to derive the soliton solution. One and two soliton solutions are generated. Using two soliton solutions of 3 CNLS equation, nonlinear tunneling of soliton for both with and without exponential background has been discussed. Finally cascade compression of optical soliton through multi-nonlinear barrier has been discussed. The obtained results may have promising applications in all-optical devices based on optical solitons, study of soliton propagation in birefringence fiber systems and optical soliton with distributed dispersion and nonlinearity management. -- Highlights: •We consider the nonlinear tunneling of soliton in birefringence fiber. •3-coupled NLS (CNLS) equation with variable coefficients is considered. •Two soliton solutions are obtained via Darboux transformation using constructed Lax pair. •Soliton tunneling through dispersion barrier and well are investigated. •Finally, cascade compression of soliton has been achieved.
Nonlinear ordinary differential equations: A discussion on symmetries and singularities
NASA Astrophysics Data System (ADS)
Paliathanasis, Andronikos; Leach, P. G. L.
2016-06-01
Two essential methods, the symmetry analysis and the singularity analysis, for the study of the integrability of nonlinear ordinary differential equations is the purpose of this work. The main similarities and the differences of these two different methods are discussed.
Oscillation theorems for second order nonlinear forced differential equations.
Salhin, Ambarka A; Din, Ummul Khair Salma; Ahmad, Rokiah Rozita; Noorani, Mohd Salmi Md
2014-01-01
In this paper, a class of second order forced nonlinear differential equation is considered and several new oscillation theorems are obtained. Our results generalize and improve those known ones in the literature. PMID:25077054
Late-time attractor for the cubic nonlinear wave equation
Szpak, Nikodem
2010-08-15
We apply our recently developed scaling technique for obtaining late-time asymptotics to the cubic nonlinear wave equation and explain the appearance and approach to the two-parameter attractor found recently by Bizon and Zenginoglu.
Multivariate Padé Approximations For Solving Nonlinear Diffusion Equations
NASA Astrophysics Data System (ADS)
Turut, V.
2015-11-01
In this paper, multivariate Padé approximation is applied to power series solutions of nonlinear diffusion equations. As it is seen from tables, multivariate Padé approximation (MPA) gives reliable solutions and numerical results.
Dispersion equation for ballooning modes in two-component plasma
NASA Astrophysics Data System (ADS)
Kozlov, D. A.; Mazur, N. G.; Pilipenko, V. A.; Fedorov, E. N.; Fedorov
2014-06-01
The ballooning magnetohydrodynamic (MHD) modes have been often suggested as a possible instability trigger of the substorm onset, and a mechanism of compressional waves in the outer magnetosphere and magnetotail. Commonly, these disturbances are characterized by the local dispersion equation that is widely applied for the description of ultra-low-frequency oscillatory disturbances and instabilities in the nightside magnetosphere. In realistic situations, especially in the inner magnetosphere, the magnetospheric plasma is composed of two components: background `cold' plasma and `hot' particles. The ballooning disturbances in a two-component plasma immersed into a curved magnetic field are described with the system of coupled equations for the Alfvén and slow magnetosonic (SMS) modes. We have reduced the basic system of MHD equations to the dispersion equation for the small-scale in transverse direction disturbances, and applied WKB approximation along a field line. As a result, we have derived a dispersion equation that can be used for geophysical applications. In particular, from this relationship the dispersion, instability threshold, and stop-bands of the Alfvén and SMS modes in two-component plasma have been examined.
Nonlinearity correction and dispersion analysis in FMCW laser radar
NASA Astrophysics Data System (ADS)
Zhao, Hao; Liu, Bingguo; Liu, Guodong; Chen, Fengdong; Zhuang, Zhitao; Yu, Yahui; Gan, Yu
2014-12-01
Frequency Modulated Continuous Wave laser radar is one of the most important ways to measure the large-size targets , combining the advantages of laser with conventional FMCW radar. Dispersion compensation and non-linear calibration are two key aspects in FMCW laser radar measurement. The paper studies the method of frequency-sampling to correct the Nonlinearity and analyzes the importance of dispersion compensation. We set up experimental verification platform, choose 1550nm band continuously tunable external cavity infrared laser as the light source, use all-fiber optical device structures, choose balanced detectors as photoelectric conversion, and finally acquire data with high speed PCI-E data acquisition card, write a measurement software with Labview. We measured the gage block 1 meter away. The experiment results show that the frequency sampling method correct the Nonlinearity well and there is a significant impact on the accuracy because of the fiber dispersion, dispersion must be compensated to obtain high accuracy. The experiment lays the foundation for further research on FMCW Laser radar.
NASA Astrophysics Data System (ADS)
Bekki, Naoaki; Shintani, Seine A.
2015-12-01
We consider the Rayleigh-Lamb-type equation for propagating pulsive waves excited by aortic-valve closure at end-systole in the human heart wall. We theoretically investigate the transcendental dispersion equation of pulsive waves for the asymmetrical zero-order mode of the Lamb wave. We analytically find a simple dispersion equation with a universal constant for a small Lamb wavenumber. We show that the simple dispersion equation can qualitatively explain the myocardial noninvasive measurements in vivo of pulsive waves in the human heart wall. We can also consistently estimate the viscoelastic constant of the myocardium in the human heart wall using the simple dispersion equation for a small Lamb wavenumber instead of using a complex nonlinear optimization.
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.
Aseeva, N. V. Gromov, E. M.; Tyutin, V. V.
2015-12-15
The dynamics of high-frequency field solitons is considered using the extended nonhomogeneous nonlinear Schrödinger equation with induced scattering from damped low-frequency waves (pseudoinduced scattering). This scattering is a 3D analog of the stimulated Raman scattering from temporal spatially homogeneous damped low-frequency modes, which is well known in optics. Spatial inhomogeneities of secondorder linear dispersion and cubic nonlinearity are also taken into account. It is shown that the shift in the 3D spectrum of soliton wavenumbers toward the short-wavelength region is due to nonlinearity increasing in coordinate and to decreasing dispersion. Analytic results are confirmed by numerical calculations.
Invariant tori for a class of nonlinear evolution equations
Kolesov, A Yu; Rozov, N Kh
2013-06-30
The paper looks at quite a wide class of nonlinear evolution equations in a Banach space, including the typical boundary value problems for the main wave equations in mathematical physics (the telegraph equation, the equation of a vibrating beam, various equations from the elastic stability and so on). For this class of equations a unified approach to the bifurcation of invariant tori of arbitrary finite dimension is put forward. Namely, the problem of the birth of such tori from the zero equilibrium is investigated under the assumption that in the stability problem for this equilibrium the situation arises close to an infinite-dimensional degeneracy. Bibliography: 28 titles.
Lattice Boltzmann model for generalized nonlinear wave equations
NASA Astrophysics Data System (ADS)
Lai, Huilin; Ma, Changfeng
2011-10-01
In this paper, a lattice Boltzmann model is developed to solve a class of the nonlinear wave equations. Through selecting equilibrium distribution function and an amending function properly, the governing evolution equation can be recovered correctly according to our proposed scheme, in which the Chapman-Enskog expansion is employed. We validate the algorithm on some problems where analytic solutions are available, including the second-order telegraph equation, the nonlinear Klein-Gordon equation, and the damped, driven sine-Gordon equation. It is found that the numerical results agree well with the analytic solutions, which indicates that the present algorithm is very effective and can be used to solve more general nonlinear problems.
Nonlinear Kramers equation associated with nonextensive statistical mechanics.
Mendes, G A; Ribeiro, M S; Mendes, R S; Lenzi, E K; Nobre, F D
2015-05-01
Stationary and time-dependent solutions of a nonlinear Kramers equation, as well as its associated nonlinear Fokker-Planck equations, are investigated within the context of Tsallis nonextensive statistical mechanics. Since no general analytical time-dependent solutions are found for such a nonlinear Kramers equation, an ansatz is considered and the corresponding asymptotic behavior is studied and compared with those known for the standard linear Kramers equation. The H-theorem is analyzed for this equation and its connection with Tsallis entropy is investigated. An application is discussed, namely the motion of Hydra cells in two-dimensional cellular aggregates, for which previous measurements have verified q-Gaussian distributions for velocity components and superdiffusion. The present analysis is in quantitative agreement with these experimental results. PMID:26066118
Solutions to Class of Linear and Nonlinear Fractional Differential Equations
NASA Astrophysics Data System (ADS)
Abdel-Salam, Emad A.-B.; Hassan, Gamal F.
2016-02-01
In this paper, the fractional auxiliary sub-equation expansion method is proposed to solve nonlinear fractional differential equations. To illustrate the effectiveness of the method, we discuss the space-time fractional KdV equation, the space-time fractional RLW equation, the space-time fractional Boussinesq equation, and the (3+1)-space-time fractional ZK equation. The solutions are expressed in terms of fractional hyperbolic and fractional trigonometric functions. These solutions are useful to understand the mechanisms of the complicated nonlinear physical phenomena and fractional differential equations. Among these solutions, some are found for the first time. The analytical solution of homogenous linear FDEs with constant coefficients are obtained by using the series and the Mittag–Leffler function methods. The obtained results recover the well-know solutions when α = 1.
Solutions to Class of Linear and Nonlinear Fractional Differential Equations
NASA Astrophysics Data System (ADS)
Emad A-B., Abdel-Salam; Gamal, F. Hassan
2016-02-01
In this paper, the fractional auxiliary sub-equation expansion method is proposed to solve nonlinear fractional differential equations. To illustrate the effectiveness of the method, we discuss the space-time fractional KdV equation, the space-time fractional RLW equation, the space-time fractional Boussinesq equation, and the (3+1)-space-time fractional ZK equation. The solutions are expressed in terms of fractional hyperbolic and fractional trigonometric functions. These solutions are useful to understand the mechanisms of the complicated nonlinear physical phenomena and fractional differential equations. Among these solutions, some are found for the first time. The analytical solution of homogenous linear FDEs with constant coefficients are obtained by using the series and the Mittag-Leffler function methods. The obtained results recover the well-know solutions when α = 1.
Nonlinear flap-lag axial equations of a rotating beam
NASA Technical Reports Server (NTRS)
Kaza, K. R. V.; Kvaternik, R. G.
1977-01-01
It is possible to identify essentially four approaches by which analysts have established either the linear or nonlinear governing equations of motion for a particular problem related to the dynamics of rotating elastic bodies. The approaches include the effective applied load artifice in combination with a variational principle and the use of Newton's second law, written as D'Alembert's principle, applied to the deformed configuration. A third approach is a variational method in which nonlinear strain-displacement relations and a first-degree displacement field are used. The method introduced by Vigneron (1975) for deriving the linear flap-lag equations of a rotating beam constitutes the fourth approach. The reported investigation shows that all four approaches make use of the geometric nonlinear theory of elasticity. An alternative method for deriving the nonlinear coupled flap-lag-axial equations of motion is also discussed.
Entropy and convexity for nonlinear partial differential equations
Ball, John M.; Chen, Gui-Qiang G.
2013-01-01
Partial differential equations are ubiquitous in almost all applications of mathematics, where they provide a natural mathematical description of many phenomena involving change in physical, chemical, biological and social processes. The concept of entropy originated in thermodynamics and statistical physics during the nineteenth century to describe the heat exchanges that occur in the thermal processes in a thermodynamic system, while the original notion of convexity is for sets and functions in mathematics. Since then, entropy and convexity have become two of the most important concepts in mathematics. In particular, nonlinear methods via entropy and convexity have been playing an increasingly important role in the analysis of nonlinear partial differential equations in recent decades. This opening article of the Theme Issue is intended to provide an introduction to entropy, convexity and related nonlinear methods for the analysis of nonlinear partial differential equations. We also provide a brief discussion about the content and contributions of the papers that make up this Theme Issue. PMID:24249768
The numerical dynamic for highly nonlinear partial differential equations
NASA Technical Reports Server (NTRS)
Lafon, A.; Yee, H. C.
1992-01-01
Problems associated with the numerical computation of highly nonlinear equations in computational fluid dynamics are set forth and analyzed in terms of the potential ranges of spurious behaviors. A reaction-convection equation with a nonlinear source term is employed to evaluate the effects related to spatial and temporal discretizations. The discretization of the source term is described according to several methods, and the various techniques are shown to have a significant effect on the stability of the spurious solutions. Traditional linearized stability analyses cannot provide the level of confidence required for accurate fluid dynamics computations, and the incorporation of nonlinear analysis is proposed. Nonlinear analysis based on nonlinear dynamical systems complements the conventional linear approach and is valuable in the analysis of hypersonic aerodynamics and combustion phenomena.
The Jeffcott equations in nonlinear rotordynamics
NASA Technical Reports Server (NTRS)
Zalik, R. A.
1987-01-01
The Jeffcott equations are a system of coupled differential equations representing the behavior of a rotating shaft. This is a simple model which allows investigation of the basic dynamic behavior of rotating machinery. Nolinearities can be introduced by taking into consideration deadband, side force, and rubbing, among others. The properties of the solutions of the Jeffcott equations with deadband are studied. In particular, it is shown how bounds for the solution of these equations can be obtained from bounds for the solutions of the linearized equations. By studying the behavior of the Fourier transforms of the solutions, we are also able to predict the onset of destructive vibrations. These conclusions are verified by means of numerical solutions of the equations, and of power spectrum density (PSD) plots. This study offers insight into a possible detection method to determine pump stability margins during flight and hot fire tests, and was motivated by the need to explain a phenomenon observed in the development phase of the cryogenic pumps of the Space Shuttle, during hot fire ground testing; namely, the appearance of vibrations at frequencies that could not be accounted for by means of linear models.
Relations between nonlinear Riccati equations and other equations in fundamental physics
NASA Astrophysics Data System (ADS)
Schuch, Dieter
2014-10-01
Many phenomena in the observable macroscopic world obey nonlinear evolution equations while the microscopic world is governed by quantum mechanics, a fundamental theory that is supposedly linear. In order to combine these two worlds in a common formalism, at least one of them must sacrifice one of its dogmas. Linearizing nonlinear dynamics would destroy the fundamental property of this theory, however, it can be shown that quantum mechanics can be reformulated in terms of nonlinear Riccati equations. In a first step, it will be shown that the information about the dynamics of quantum systems with analytical solutions can not only be obtainable from the time-dependent Schrödinger equation but equally-well from a complex Riccati equation. Comparison with supersymmetric quantum mechanics shows that even additional information can be obtained from the nonlinear formulation. Furthermore, the time-independent Schrödinger equation can also be rewritten as a complex Riccati equation for any potential. Extension of the Riccati formulation to include irreversible dissipative effects is straightforward. Via (real and complex) Riccati equations, other fields of physics can also be treated within the same formalism, e.g., statistical thermodynamics, nonlinear dynamical systems like those obeying a logistic equation as well as wave equations in classical optics, Bose- Einstein condensates and cosmological models. Finally, the link to abstract "quantizations" such as the Pythagorean triples and Riccati equations connected with trigonometric and hyperbolic functions will be shown.
Nonlinear acoustic pulse propagation in dispersive sediments using fractional loss operators.
Maestas, Joseph T; Collis, Jon M
2016-03-01
The nonlinear progressive wave equation (NPE) is a time-domain formulation of the Euler fluid equations designed to model low-angle wave propagation using a wave-following computational domain. The wave-following frame of reference permits the simulation of long-range propagation and is useful in modeling blast wave effects in the ocean waveguide. Existing models do not take into account frequency-dependent sediment attenuation, a feature necessary for accurately describing sound propagation over, into, and out of the ocean sediment. Sediment attenuation is addressed in this work by applying lossy operators to the governing equation that are based on a fractional Laplacian. These operators accurately describe frequency-dependent attenuation and dispersion in typical ocean sediments. However, dispersion within the sediment is found to be a secondary process to absorption and effectively negligible for ranges of interest. The resulting fractional NPE is benchmarked against a Fourier-transformed parabolic equation solution for a linear case, and against the analytical Mendousse solution to Burgers' equation for the nonlinear case. The fractional NPE is then used to investigate the effects of attenuation on shock wave propagation. PMID:27036279
Generalized nonlinear Proca equation and its free-particle solutions
NASA Astrophysics Data System (ADS)
Nobre, F. D.; Plastino, A. R.
2016-06-01
We introduce a nonlinear extension of Proca's field theory for massive vector (spin 1) bosons. The associated relativistic nonlinear wave equation is related to recently advanced nonlinear extensions of the Schrödinger, Dirac, and Klein-Gordon equations inspired on the non-extensive generalized thermostatistics. This is a theoretical framework that has been applied in recent years to several problems in nuclear and particle physics, gravitational physics, and quantum field theory. The nonlinear Proca equation investigated here has a power-law nonlinearity characterized by a real parameter q (formally corresponding to the Tsallis entropic parameter) in such a way that the standard linear Proca wave equation is recovered in the limit q → 1. We derive the nonlinear Proca equation from a Lagrangian, which, besides the usual vectorial field Ψ ^{μ }(ěc {x},t), involves an additional field Φ ^{μ }(ěc {x},t). We obtain exact time-dependent soliton-like solutions for these fields having the form of a q-plane wave, and we show that both field equations lead to the relativistic energy-momentum relation E2 = p2c2 + m2c4 for all values of q. This suggests that the present nonlinear theory constitutes a new field theoretical representation of particle dynamics. In the limit of massless particles the present q-generalized Proca theory reduces to Maxwell electromagnetism, and the q-plane waves yield localized, transverse solutions of Maxwell equations. Physical consequences and possible applications are discussed.
Optimal Variational Asymptotic Method for Nonlinear Fractional Partial Differential Equations
Baranwal, Vipul K.; Pandey, Ram K.
2014-01-01
We propose optimal variational asymptotic method to solve time fractional nonlinear partial differential equations. In the proposed method, an arbitrary number of auxiliary parameters γ0, γ1, γ2,… and auxiliary functions H0(x), H1(x), H2(x),… are introduced in the correction functional of the standard variational iteration method. The optimal values of these parameters are obtained by minimizing the square residual error. To test the method, we apply it to solve two important classes of nonlinear partial differential equations: (1) the fractional advection-diffusion equation with nonlinear source term and (2) the fractional Swift-Hohenberg equation. Only few iterations are required to achieve fairly accurate solutions of both the first and second problems.
An iterative method for systems of nonlinear hyperbolic equations
NASA Technical Reports Server (NTRS)
Scroggs, Jeffrey S.
1989-01-01
An iterative algorithm for the efficient solution of systems of nonlinear hyperbolic equations is presented. Parallelism is evident at several levels. In the formation of the iteration, the equations are decoupled, thereby providing large grain parallelism. Parallelism may also be exploited within the solves for each equation. Convergence of the interation is established via a bounding function argument. Experimental results in two-dimensions are presented.
Liapunov functions for non-linear difference equation stability analysis.
NASA Technical Reports Server (NTRS)
Park, K. E.; Kinnen, E.
1972-01-01
Liapunov functions to determine the stability of non-linear autonomous difference equations can be developed through the use of auxiliary exact difference equations. For this purpose definitions are introduced for the gradient of an implicit function of a discrete variable, a principal sum, a definite sum and an exact difference equation, and a theorem for exactness of a difference form is proved. Examples illustrate the procedure.
Global well-posedness for the 2D dispersive SQG equation and inviscid Boussinesq equations
NASA Astrophysics Data System (ADS)
Wan, Renhui; Chen, Jiecheng
2016-08-01
In this paper, we obtain global well-posedness for the 2D dispersive SQG equation and inviscid Boussinesq equations. Our works are consistent with the corresponding works by Elgindi-Widmayer (SIAM J Math Anal 47:4672-4684, 2015) in the special case {A=κ=1}. In addition, our result concerning the SQG equation can be regarded as the borderline case of the work by Cannone et al. (Proc Lond Math Soc 106:650-674, 2013).
Nonlinear Resonance and Duffing's Spring Equation
ERIC Educational Resources Information Center
Fay, Temple H.
2006-01-01
This note discusses the boundary in the frequency--amplitude plane for boundedness of solutions to the forced spring Duffing type equation. For fixed initial conditions and fixed parameter [epsilon] results are reported of a systematic numerical investigation on the global stability of solutions to the initial value problem as the parameters F and…
Nonlinear Resonance and Duffing's Spring Equation II
ERIC Educational Resources Information Center
Fay, T. H.; Joubert, Stephan V.
2007-01-01
The paper discusses the boundary in the frequency-amplitude plane for boundedness of solutions to the forced spring Duffing type equation x[umlaut] + x + [epsilon]x[cubed] = F cos[omega]t. For fixed initial conditions and for representative fixed values of the parameter [epsilon], the results are reported of a systematic numerical investigation…
Non-Linear Spring Equations and Stability
ERIC Educational Resources Information Center
Fay, Temple H.; Joubert, Stephan V.
2009-01-01
We discuss the boundary in the Poincare phase plane for boundedness of solutions to spring model equations of the form [second derivative of]x + x + epsilonx[superscript 2] = Fcoswt and the [second derivative of]x + x + epsilonx[superscript 3] = Fcoswt and report the results of a systematic numerical investigation on the global stability of…
A Procedure to Construct Conservation Laws of Nonlinear Evolution Equations
NASA Astrophysics Data System (ADS)
Yaşar, Emrullah; San, Sait
2016-05-01
In this article, we established abundant local conservation laws to some nonlinear evolution equations by a new combined approach, which is a union of multiplier and Ibragimov's new conservation theorem method. One can conclude that the solutions of the adjoint equations corresponding to the new conservation theorem can be obtained via multiplier functions. Many new families of conservation laws of the Pochammer-Chree (PC) equation and the Kaup-Boussinesq type of coupled KdV system are successfully obtained. The combined method presents a wider applicability for handling the conservation laws of nonlinear wave equations. The conserved vectors obtained here can be important for the explanation of some practical physical problems, reductions, and solutions of the underlying equations.
NASA Astrophysics Data System (ADS)
Baskonus, Haci Mehmet; Bulut, Hasan
2015-10-01
In this paper, a new computational algorithm called the "Improved Bernoulli sub-equation function method" has been proposed. This algorithm is based on the Bernoulli Sub-ODE method. Firstly, the nonlinear evaluation equations used for representing various physical phenomena are converted into ordinary differential equations by using various wave transformations. In this way, nonlinearity is preserved and represent nonlinear physical problems. The nonlinearity of physical problems together with the derivations is seen as the secret key to solve the general structure of problems. The proposed analytical schema, which is newly submitted to the literature, has been expressed comprehensively in this paper. The analytical solutions, application results, and comparisons are presented by plotting the two and three dimensional surfaces of analytical solutions obtained by using the methods proposed for some important nonlinear physical problems. Finally, a conclusion has been presented by mentioning the important discoveries in this study.
Algebraic calculation of stroboscopic maps of ordinary, nonlinear differential equations
Wackerbauer, R. ); Huebler, A. . Center for Complex Systems Research); Mayer-Kress, G. California Univ., Santa Cruz, CA . Dept. of Mathematics)
1991-07-25
The relation between the parameters of a differential equation and corresponding discrete maps are becoming increasingly important in the study of nonlinear dynamical systems. Maps are well adopted for numerical computation and several universal properties of them are known. Therefore some perturbation methods have been proposed to deduce them for physical systems, which can be modeled by an ordinary differential equation (ODE) with a small nonlinearity. A new iterative, rigorous algebraic method for the calculation of the coefficients of a Taylor expansion of a stroboscopic map from ODE's with not necessarily small nonlinearities is presented. It is shown analytically that most of the coefficients are small for a small integration time and grow slowly in the course of time if the flow vector field of the ODE is polynomial and if the ODE has fixed point in the origin. Approximations of different orders respectively of the rest term are investigated for several nonlinear systems. 31 refs., 16 figs.
A Jacobi collocation approximation for nonlinear coupled viscous Burgers' equation
NASA Astrophysics Data System (ADS)
Doha, Eid; Bhrawy, Ali; Abdelkawy, Mohamed; Hafez, Ramy
2014-02-01
This article presents a numerical approximation of the initial-boundary nonlinear coupled viscous Burgers' equation based on spectral methods. A Jacobi-Gauss-Lobatto collocation (J-GL-C) scheme in combination with the implicit Runge-Kutta-Nyström (IRKN) scheme are employed to obtain highly accurate approximations to the mentioned problem. This J-GL-C method, based on Jacobi polynomials and Gauss-Lobatto quadrature integration, reduces solving the nonlinear coupled viscous Burgers' equation to a system of nonlinear ordinary differential equation which is far easier to solve. The given examples show, by selecting relatively few J-GL-C points, the accuracy of the approximations and the utility of the approach over other analytical or numerical methods. The illustrative examples demonstrate the accuracy, efficiency, and versatility of the proposed algorithm.
NASA Astrophysics Data System (ADS)
Tchinang Tchameu, J. D.; Togueu Motcheyo, A. B.; Tchawoua, C.
2016-09-01
The discrete multi-rogue waves (DMRW) as solution of the discrete nonlinear Schrödinger (DNLS) equation with saturable nonlinearities is studied numerically. These biological rogue waves represent the complex probability amplitude of finding an amide-I vibrational quantum at a site. We observe that the growth in the higher order saturable nonlinearity implies the formation of DMRW including an increase in the short-living DMRW and a decrease in amplitude of the long-living DMRW.
Stochasticity in numerical solutions of the nonlinear Schroedinger equation
NASA Technical Reports Server (NTRS)
Shen, Mei-Mei; Nicholson, D. R.
1987-01-01
The cubically nonlinear Schroedinger equation is an important model of nonlinear phenomena in fluids and plasmas. Numerical solutions in a spatially periodic system commonly involve truncation to a finite number of Fourier modes. These solutions are found to be stochastic in the sense that the largest Liapunov exponent is positive. As the number of modes is increased, the size of this exponent appears to converge to zero, in agreement with the recent demonstration of the integrability of the spatially periodic case.
The Buoyancy Budget With a Nonlinear Equation of State
NASA Astrophysics Data System (ADS)
Hieronymus, M. H.; Nycander, J.
2012-12-01
There has been a number of studies focusing on different aspects of having a nonlinear equation of state for seawater. Amongst other things it has been shown that the nonlinear equation of state has implications for the oceanic energy budget and that nonlinear processes can be a significant source of dense water production. This presentation will focus on the oceanic buoyancy budget. The nonlinear equation of state of seawater can introduce a sink or source of buoyancy when water parcels of unequal salinities and temperatures are mixed. A common example is the process known as cabbeling, which is responsible for forming a water mass that is denser than the original constituents in a mixture of two water masses with equal densities but different salinities and temperatures. This presentation will contain quantitative estimates of these nonlinear effects on the buoyancy budget of the global ocean. Because of these nonlinear effects there is a net sink of buoyancy in the oceans interior and the size of this sink can be determined from the buoyancy fluxes at the ocean boundaries. These boundary buoyancy fluxes are calculated using two surface heat flux climatologies one based on in situ measurements, the other on a reanalysis and in both cases using a nonlinear equation of state. The presentation also treats the buoyancy budget in the State of the art ocean model Nucleus for European Modelling of the Ocean (NEMO) and the results from NEMO are seen to be in good agreement with the buoyancy budgets based on the heat flux climatologies. Using the ocean model is a good complement to the surface flux climatologies, because in NEMO the buoyancy fluxes can be evaluated at all vertical model levels. This means that the vertical distribution of the buoyancy sink can be looked into. The results from NEMO shows that in large parts of the ocean the nonlinear buoyancy sink is the largest contribution to the buoyancy budget.
Nonlinear generalized master equations and accounting for initial correlations
NASA Astrophysics Data System (ADS)
Los, V. F.
2009-08-01
We develop a new method based on using a time-dependent operator (generally not a projection operator) converting a distribution function (statistical operator) of a total system into the relevant form that allows deriving new exact nonlinear generalized master equations (GMEs). The derived inhomogeneous nonlinear GME is a generalization of the linear Nakajima-Zwanzig GME and can be viewed as an alternative to the BBGKY chain. It is suitable for obtaining both nonlinear and linear evolution equations. As in the conventional linear GME, there is an inhomogeneous term comprising all multiparticle initial correlations. To include the initial correlations into consideration, we convert the obtained inhomogeneous nonlinear GME into the homogenous form by the previously suggested method. We use no conventional approximation like the random phase approximation (RPA) or the Bogoliubov principle of weakening of initial correlations. The obtained exact homogeneous nonlinear GME describes all evolution stages of the (sub)system of interest and treats initial correlations on an equal footing with collisions via the modified memory kernel. As an application, we obtain a new homogeneous nonlinear equation retaining initial correlations for a one-particle distribution function of the spatially inhomogeneous nonideal gas of classical particles. In contrast to existing approaches, this equation holds for all time scales and takes the influence of pair collisions and initial correlations on the dissipative and nondissipative characteristics of the system into account consistently with the adopted approximation (linear in the gas density). We show that on the kinetic time scale, the time-reversible terms resulting from the initial correlations vanish (if the particle dynamics are endowed with the mixing property) and this equation can be converted into the Vlasov-Landau and Boltzmann equations without any additional commonly used approximations. The entire process of transition can
Quantification and prediction of extreme events in a one-dimensional nonlinear dispersive wave model
NASA Astrophysics Data System (ADS)
Cousins, Will; Sapsis, Themistoklis P.
2014-07-01
The aim of this work is the quantification and prediction of rare events characterized by extreme intensity in nonlinear waves with broad spectra. We consider a one-dimensional nonlinear model with deep-water waves dispersion relation, the Majda-McLaughlin-Tabak (MMT) model, in a dynamical regime that is characterized by a broadband spectrum and strong nonlinear energy transfers during the development of intermittent events with finite-lifetime. To understand the energy transfers that occur during the development of an extreme event we perform a spatially localized analysis of the energy distribution along different wavenumbers by means of the Gabor transform. A statistical analysis of the Gabor coefficients reveals (i) the low-dimensionality of the intermittent structures, (ii) the interplay between non-Gaussian statistical properties and nonlinear energy transfers between modes, as well as (iii) the critical scales (or critical Gabor coefficients) where a critical amount of energy can trigger the formation of an extreme event. We analyze the unstable character of these special localized modes directly through the system equation and show that these intermittent events are due to the interplay of the system nonlinearity, the wave dispersion, and the wave dissipation which mimics wave breaking. These localized instabilities are triggered by random localizations of energy in space, created by the dispersive propagation of low-amplitude waves with random phase. Based on these properties, we design low-dimensional functionals of these Gabor coefficients that allow for the prediction of the extreme event well before the nonlinear interactions begin to occur.
Decay and stability for nonlinear hyperbolic equations
NASA Astrophysics Data System (ADS)
Marcati, Pierangelo
This paper deals with the asymptotic stability of the null solution of a semilinear partial differential equation. The La Salle Invariance Principle has been used to obtain the stability results. The first result is given under quite general hypotheses assuming only the precompactness of the orbits and the local existence. In the second part, under some restrictions, sufficient conditions for precompactness of the orbits and decay of solutions are given. An existence and uniqueness theorem is proved in the Appendix. Some examples are given.
Combined optical solitons with parabolic law nonlinearity and spatio-temporal dispersion
NASA Astrophysics Data System (ADS)
Zhou, Qin; Zhu, Qiuping
2015-03-01
In this work, combined optical solitons are constructed in a weakly nonlocal nonlinear medium. The spatio-temporal dispersion (STD), parabolic law nonlinearity, detuning, nonlinear dispersion as well as inter-modal dispersion are taken into account. The integration tool that is applied is the complex envelope function ansatz. The influences of different parameters on dynamical behavior of combined optical solitons are discussed. The results are useful in describing the propagation of combined optical solitons with STD and parabolic law nonlinearity.
Cylindrical nonlinear Schroedinger equation versus cylindrical Korteweg-de Vries equation
Fedele, Renato; De Nicola, Sergio; Grecu, Dan; Visinescu, Anca; Shukla, Padma K.
2008-10-15
A correspondence between the family of cylindrical nonlinear Schroedinger (cNLS) equations and the one of cylindrical Korteweg-de Vries (cKdV) equations is constructed. It associates non stationary solutions of the first family with the ones of the second family. This is done by using a correspondence, recently found, between the families of generalized NLS equation and generalized KdV equation, and their solutions in the form of travelling waves, respectively. In particular, non-stationary soliton-like solutions of the cNLS equation can be associated with non-stationary soliton-like solutions of cKdV equation.
NASA Astrophysics Data System (ADS)
Yang, Xiao; Du, Dianlou
2010-08-01
The Poisson structure on CN×RN is introduced to give the Hamiltonian system associated with a spectral problem which yields the nonlinear Schrödinger (NLS) hierarchy. The Hamiltonian system is proven to be Liouville integrable. Some (2+1)-dimensional equations including NLS equation, Kadomtesev-Petviashvili I (KPI) equation, coupled KPI equation, and modified Kadomtesev-Petviashvili (mKP) equation, are decomposed into Hamilton flows via the NLS hierarchy. The algebraic curve, Abel-Jacobi coordinates, and Riemann-Jacobi inversion are used to obtain the algebrogeometric solutions of these equations.
Yang Xiao; Du Dianlou
2010-08-15
The Poisson structure on C{sup N}xR{sup N} is introduced to give the Hamiltonian system associated with a spectral problem which yields the nonlinear Schroedinger (NLS) hierarchy. The Hamiltonian system is proven to be Liouville integrable. Some (2+1)-dimensional equations including NLS equation, Kadomtesev-Petviashvili I (KPI) equation, coupled KPI equation, and modified Kadomtesev-Petviashvili (mKP) equation, are decomposed into Hamilton flows via the NLS hierarchy. The algebraic curve, Abel-Jacobi coordinates, and Riemann-Jacobi inversion are used to obtain the algebrogeometric solutions of these equations.
NASA Technical Reports Server (NTRS)
Rosen, I. G.
1988-01-01
An approximation and convergence theory was developed for Galerkin approximations to infinite dimensional operator Riccati differential equations formulated in the space of Hilbert-Schmidt operators on a separable Hilbert space. The Riccati equation was treated as a nonlinear evolution equation with dynamics described by a nonlinear monotone perturbation of a strongly coercive linear operator. A generic approximation result was proven for quasi-autonomous nonlinear evolution system involving accretive operators which was then used to demonstrate the Hilbert-Schmidt norm convergence of Galerkin approximations to the solution of the Riccati equation. The application of the results was illustrated in the context of a linear quadratic optimal control problem for a one dimensional heat equation.
Rogue waves for a system of coupled derivative nonlinear Schrödinger equations
NASA Astrophysics Data System (ADS)
Chan, H. N.; Malomed, B. A.; Chow, K. W.; Ding, E.
2016-01-01
Rogue waves (RWs) are unexpectedly strong excitations emerging from an otherwise tranquil background. The nonlinear Schrödinger equation (NLSE), a ubiquitous model with wide applications to fluid mechanics, optics, plasmas, etc., exhibits RWs only in the regime of modulation instability (MI) of the background. For a system of multiple waveguides, the governing coupled NLSEs can produce regimes of MI and RWs, even if each component has dispersion and cubic nonlinearity of opposite signs. A similar effect is demonstrated here for a system of coupled derivative NLSEs (DNLSEs) where the special feature is the nonlinear self-steepening of narrow pulses. More precisely, these additional regimes of MI and RWs for coupled DNLSEs depend on the mismatch in group velocities between the components, and the parameters for cubic nonlinearity and self-steepening. RWs considered in this paper differ from those of the NLSEs in terms of the amplification ratio and criteria of existence. Applications to optics and plasma physics are discussed.
On the Dirichlet problem for a nonlinear elliptic equation
NASA Astrophysics Data System (ADS)
Egorov, Yu V.
2015-04-01
We prove the existence of an infinite set of solutions to the Dirichlet problem for a nonlinear elliptic equation of the second order. Such a problem for a nonlinear elliptic equation with Laplace operator was studied earlier by Krasnosel'skii, Bahri, Berestycki, Lions, Rabinowitz, Struwe and others. We study the spectrum of this problem and prove the weak convergence to 0 of the sequence of normed eigenfunctions. Moreover, we obtain some estimates for the 'Fourier coefficients' of functions in W^1p,0(Ω). This allows us to improve the preceding results. Bibliography: 8 titles.
Nonlinear dispersion of a pollutant ejected into a channel flow
NASA Astrophysics Data System (ADS)
Van Gorder, Robert A.; Vajravelu, Kuppalapalle
2011-10-01
In this paper, we study the nonlinear coupled boundary value problem arising from the nonlinear dispersion of a pollutant ejected by an external source into a channel flow. We obtain exact solutions for the steady flow for some special cases and an implicit exact solution for the unsteady flow. Additionally, we obtain analytical solutions for the transient flow. From the obtained solutions, we are able to deduce the qualitative influence of the model parameters on the solutions. Furthermore, we are able to give both exact and analytical expressions for the skin friction and wall mass transfer rate as functions of the model parameters. The model considered can be useful for understanding the polluting situations of an improper discharge incident and evaluating the effects of decontaminating measures for the water bodies.
Zhao Dun; Zhang Yujuan; Lou Weiwei; Luo Honggang
2011-04-15
By constructing nonisospectral Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy, we investigate the nonautonomous nonlinear Schroedinger (NLS) equations which have been used to describe the Feshbach resonance management in matter-wave solitons in Bose-Einstein condensate and the dispersion and nonlinearity managements for optical solitons. It is found that these equations are some special cases of a new integrable model of nonlocal nonautonomous NLS equations. Based on the Lax pairs, the Darboux transformation and conservation laws are explored. It is shown that the local external potentials would break down the classical infinite number of conservation laws. The result indicates that the integrability of the nonautonomous NLS systems may be nontrivial in comparison to the conventional concept of integrability in the canonical case.
NASA Astrophysics Data System (ADS)
Nikitenkova, S.; Singh, N.; Stepanyants, Y.
2015-12-01
In this paper, we revisit the problem of modulation stability of quasi-monochromatic wave-trains propagating in a media with the double dispersion occurring both at small and large wavenumbers. We start with the shallow-water equations derived by Shrira [Izv., Acad. Sci., USSR, Atmos. Ocean. Phys. (Engl. Transl.) 17, 55-59 (1981)] which describes both surface and internal long waves in a rotating fluid. The small-scale (Boussinesq-type) dispersion is assumed to be weak, whereas the large-scale (Coriolis-type) dispersion is considered as without any restriction. For unidirectional waves propagating in one direction, only the considered set of equations reduces to the Gardner-Ostrovsky equation which is applicable only within a finite range of wavenumbers. We derive the nonlinear Schrödinger equation (NLSE) which describes the evolution of narrow-band wave-trains and show that within a more general bi-directional equation the wave-trains, similar to that derived from the Ostrovsky equation, are also modulationally stable at relatively small wavenumbers k < kc and unstable at k > kc, where kc is some critical wavenumber. The NLSE derived here has a wider range of applicability: it is valid for arbitrarily small wavenumbers. We present the analysis of coefficients of the NLSE for different signs of coefficients of the governing equation and compare them with those derived from the Ostrovsky equation. The analysis shows that for weakly dispersive waves in the range of parameters where the Gardner-Ostrovsky equation is valid, the cubic nonlinearity does not contribute to the nonlinear coefficient of NLSE; therefore, the NLSE can be correctly derived from the Ostrovsky equation.
Nikitenkova, S; Singh, N; Stepanyants, Y
2015-12-01
In this paper, we revisit the problem of modulation stability of quasi-monochromatic wave-trains propagating in a media with the double dispersion occurring both at small and large wavenumbers. We start with the shallow-water equations derived by Shrira [Izv., Acad. Sci., USSR, Atmos. Ocean. Phys. (Engl. Transl.) 17, 55-59 (1981)] which describes both surface and internal long waves in a rotating fluid. The small-scale (Boussinesq-type) dispersion is assumed to be weak, whereas the large-scale (Coriolis-type) dispersion is considered as without any restriction. For unidirectional waves propagating in one direction, only the considered set of equations reduces to the Gardner-Ostrovsky equation which is applicable only within a finite range of wavenumbers. We derive the nonlinear Schrödinger equation (NLSE) which describes the evolution of narrow-band wave-trains and show that within a more general bi-directional equation the wave-trains, similar to that derived from the Ostrovsky equation, are also modulationally stable at relatively small wavenumbers k < kc and unstable at k > kc, where kc is some critical wavenumber. The NLSE derived here has a wider range of applicability: it is valid for arbitrarily small wavenumbers. We present the analysis of coefficients of the NLSE for different signs of coefficients of the governing equation and compare them with those derived from the Ostrovsky equation. The analysis shows that for weakly dispersive waves in the range of parameters where the Gardner-Ostrovsky equation is valid, the cubic nonlinearity does not contribute to the nonlinear coefficient of NLSE; therefore, the NLSE can be correctly derived from the Ostrovsky equation. PMID:26723152
Multiply scaled constrained nonlinear equation solvers. [for nonlinear heat conduction problems
NASA Technical Reports Server (NTRS)
Padovan, Joe; Krishna, Lala
1986-01-01
To improve the numerical stability of nonlinear equation solvers, a partitioned multiply scaled constraint scheme is developed. This scheme enables hierarchical levels of control for nonlinear equation solvers. To complement the procedure, partitioned convergence checks are established along with self-adaptive partitioning schemes. Overall, such procedures greatly enhance the numerical stability of the original solvers. To demonstrate and motivate the development of the scheme, the problem of nonlinear heat conduction is considered. In this context the main emphasis is given to successive substitution-type schemes. To verify the improved numerical characteristics associated with partitioned multiply scaled solvers, results are presented for several benchmark examples.
Conservation laws of inviscid Burgers equation with nonlinear damping
NASA Astrophysics Data System (ADS)
Abdulwahhab, Muhammad Alim
2014-06-01
In this paper, the new conservation theorem presented in Ibragimov (2007) [14] is used to find conservation laws of the inviscid Burgers equation with nonlinear damping ut+g(u)ux+λh(u)=0. We show that this equation is both quasi self-adjoint and self-adjoint, and use these concepts to simplify conserved quantities for various choices of g(u) and h(u).
Intermittency and solitons in the driven dissipative nonlinear Schroedinger equation
NASA Technical Reports Server (NTRS)
Moon, H. T.; Goldman, M. V.
1984-01-01
The cubic nonlinear Schroedinger equation, in the presence of driving and Landau damping, is studied numerically. As the pump intensity is increased, the system exhibits a transition from intermittency to a two-torus to chaos. The laminar phase of the intermittency is also a two-torus motion which corresponds in physical space to two identical solitons of amplitude determined by a power-balance equation.
Optimization of a finite difference method for nonlinear wave equations
NASA Astrophysics Data System (ADS)
Chen, Miaochao
2013-07-01
Wave equations have important fluid dynamics background, which are extensively used in many fields, such as aviation, meteorology, maritime, water conservancy, etc. This paper is devoted to the explicit difference method for nonlinear wave equations. Firstly, a three-level and explicit difference scheme is derived. It is shown that the explicit difference scheme is uniquely solvable and convergent. Moreover, a numerical experiment is conducted to illustrate the theoretical results of the presented method.
Dam break problem for the focusing nonlinear Schrödinger equation and the generation of rogue waves
NASA Astrophysics Data System (ADS)
El, G. A.; Khamis, E. G.; Tovbis, A.
2016-09-01
We propose a novel, analytically tractable, scenario of the rogue wave formation in the framework of the small-dispersion focusing nonlinear Schrödinger (NLS) equation with the initial condition in the form of a rectangular barrier (a ‘box’). We use the Whitham modulation theory combined with the nonlinear steepest descent for the semi-classical inverse scattering transform, to describe the evolution and interaction of two counter-propagating nonlinear wave trains—the dispersive dam break flows—generated in the NLS box problem. We show that the interaction dynamics results in the emergence of modulated large-amplitude quasi-periodic breather lattices whose amplitude profiles are closely approximated by the Akhmediev and Peregrine breathers within certain space-time domain. Our semi-classical analytical results are shown to be in excellent agreement with the results of direct numerical simulations of the small-dispersion focusing NLS equation.
NASA Astrophysics Data System (ADS)
Kanka, Jiri
2009-05-01
For most telecom nonlinear applications a high effective nonlinearity, low group velocity dispersion with a low dispersion slope and a short fibre length are the key parameters. Combining photonic crystal fibre (PCF) technology with highly nonlinear glasses could meet these requirements very well. We have performed dispersion optimization of PCFs made from selected nonlinear glasses with a solid core and small number of hexagonally arrayed air holes. The optimization procedure employs the Nelder-Mead downhill simplex algorithm. For the modal analysis of the photonic crystal fibre structure a fully-vectorial mode solver based on the finite element method is used. We have obtained two types of dispersion optimized nonlinear PCF designs: PCFs of the first type are single-mode and highly nonlinear with a small and flattened dispersion in the 1500-1600 nm range. These PCF structures have air holes hexagonally arrayed in from 3 to 5 rings, however, their dispersion characteristics are very sensitive to variations in structural parameters. PCFs of the second type are two-ring PCFs with larger multi-mode cores. They have fundamental mode's zero dispersion wavelength around 1550 nm with non-zero moderate dispersion slopes which are less sensitive to structural variation. It is supposed that this alternative PCF design will be easier to fabricate. The effects of fabrication imprecision on the dispersion characteristics for both PCF designs are demonstrated numerically and discussed in the context of nonlinear telecom applications.
Maximum Likelihood Estimation of Nonlinear Structural Equation Models.
ERIC Educational Resources Information Center
Lee, Sik-Yum; Zhu, Hong-Tu
2002-01-01
Developed an EM type algorithm for maximum likelihood estimation of a general nonlinear structural equation model in which the E-step is completed by a Metropolis-Hastings algorithm. Illustrated the methodology with results from a simulation study and two real examples using data from previous studies. (SLD)
Model Comparison of Nonlinear Structural Equation Models with Fixed Covariates.
ERIC Educational Resources Information Center
Lee, Sik-Yum; Song, Xin-Yuan
2003-01-01
Proposed a new nonlinear structural equation model with fixed covariates to deal with some complicated substantive theory and developed a Bayesian path sampling procedure for model comparison. Illustrated the approach with an illustrative example using data from an international study. (SLD)
Local Influence Analysis of Nonlinear Structural Equation Models
ERIC Educational Resources Information Center
Lee, Sik-Yum; Tang, Nian-Sheng
2004-01-01
By regarding the latent random vectors as hypothetical missing data and based on the conditional expectation of the complete-data log-likelihood function in the EM algorithm, we investigate assessment of local influence of various perturbation schemes in a nonlinear structural equation model. The basic building blocks of local influence analysis…
Topological horseshoes in travelling waves of discretized nonlinear wave equations
Chen, Yi-Chiuan; Chen, Shyan-Shiou; Yuan, Juan-Ming
2014-04-15
Applying the concept of anti-integrable limit to coupled map lattices originated from space-time discretized nonlinear wave equations, we show that there exist topological horseshoes in the phase space formed by the initial states of travelling wave solutions. In particular, the coupled map lattices display spatio-temporal chaos on the horseshoes.
Tensor methods for large sparse systems of nonlinear equations
Bouaricha, A.; Schnabel, R.B.
1996-12-31
This paper introduces censor methods for solving, large sparse systems of nonlinear equations. Tensor methods for nonlinear equations were developed in the context of solving small to medium- sized dense problems. They base each iteration on a quadratic model of the nonlinear equations. where the second-order term is selected so that the model requires no more derivative or function information per iteration than standard linear model-based methods, and hardly more storage or arithmetic operations per iteration. Computational experiments on small to medium-sized problems have shown censor methods to be considerably more efficient than standard Newton-based methods, with a particularly large advantage on singular problems. This paper considers the extension of this approach to solve large sparse problems. The key issue that must be considered is how to make efficient use of sparsity in forming and solving the censor model problem at each iteration. Accomplishing this turns out to require an entirely new way of solving the tensor model that successfully exploits the sparsity of the Jacobian, whether the Jacobian is nonsingular or singular. We develop such an approach and, based upon it, an efficient tensor method for solving large sparse systems of nonlinear equations. Test results indicate that this tensor method is significantly more efficient and robust than an efficient sparse Newton-based method. in terms of iterations, function evaluations. and execution time.
Bounded and periodic solutions of nonlinear functional differential equations
Slyusarchuk, Vasilii E
2012-05-31
Conditions for the existence of bounded and periodic solutions of the nonlinear functional differential equation d{sup m}x(t)/dt{sup m} + (Fx)(t) = h(t), t element of R, are presented, involving local linear approximations to the operator F. Bibliography: 23 titles.
Painleve analysis for a nonlinear Schroedinger equation in three dimensions
Chowdhury, A.R.; Chanda, P.K.
1987-09-01
A Painleve analysis is performed for the nonlinear Schroedinger equation in (2 + 1) dimensions following the methodology of Weiss et al. simplified in the sense of Kruskal. At least for one branch it is found that the required number of arbitrary functions (as demanded by the Cauchy-Kovalevskaya theorem) exists, signalling complete integrability.
An Efficient Numerical Approach for Nonlinear Fokker-Planck equations
NASA Astrophysics Data System (ADS)
Otten, Dustin; Vedula, Prakash
2009-03-01
Fokker-Planck equations which are nonlinear with respect to their probability densities that occur in many nonequilibrium systems relevant to mean field interaction models, plasmas, classical fermions and bosons can be challenging to solve numerically. To address some underlying challenges in obtaining numerical solutions, we propose a quadrature based moment method for efficient and accurate determination of transient (and stationary) solutions of nonlinear Fokker-Planck equations. In this approach the distribution function is represented as a collection of Dirac delta functions with corresponding quadrature weights and locations, that are in turn determined from constraints based on evolution of generalized moments. Properties of the distribution function can be obtained by solution of transport equations for quadrature weights and locations. We will apply this computational approach to study a wide range of problems, including the Desai-Zwanzig Model (for nonlinear muscular contraction) and multivariate nonlinear Fokker-Planck equations describing classical fermions and bosons, and will also demonstrate good agreement with results obtained from Monte Carlo and other standard numerical methods.
Forced oscillations of nonlinear damped equation of suspended string
NASA Astrophysics Data System (ADS)
Yamaguchi, Masaru; Nagai, Tohru; Matsukane, Katsuya
2008-06-01
We shall study the existence of time-periodic solutions of nonlinear damped equation of suspended string to which a periodic nonlinear force works. We shall be conterned with weak, strong and classical time-periodic solutions and also the regularity of the solutions. To formulate our results, we shall take suitable weighted Sobolev-type spaces introduced by [M. Yamaguchi, Almost periodic oscillations of suspended string under quasiperiodic linear force, J. Math. Anal. Appl. 303 (2) (2005) 643-660; M. Yamaguchi, Infinitely many time-periodic solutions of nonlinear equation of suspended string, Funkcial. Ekvac., in press]. We shall study properties of the function spaces and show inequalities on the function spaces. To show our results we shall apply the Schauder fixed point theorem and the fixed point continuation theorem in the function spaces.
Optimal analytic method for the nonlinear Hasegawa-Mima equation
NASA Astrophysics Data System (ADS)
Baxter, Mathew; Van Gorder, Robert A.; Vajravelu, Kuppalapalle
2014-05-01
The Hasegawa-Mima equation is a nonlinear partial differential equation that describes the electric potential due to a drift wave in a plasma. In the present paper, we apply the method of homotopy analysis to a slightly more general Hasegawa-Mima equation, which accounts for hyper-viscous damping or viscous dissipation. First, we outline the method for the general initial/boundary value problem over a compact rectangular spatial domain. We use a two-stage method, where both the convergence control parameter and the auxiliary linear operator are optimally selected to minimize the residual error due to the approximation. To do the latter, we consider a family of operators parameterized by a constant which gives the decay rate of the solutions. After outlining the general method, we consider a number of concrete examples in order to demonstrate the utility of this approach. The results enable us to study properties of the initial/boundary value problem for the generalized Hasegawa-Mima equation. In several cases considered, we are able to obtain solutions with extremely small residual errors after relatively few iterations are computed (residual errors on the order of 10-15 are found in multiple cases after only three iterations). The results demonstrate that selecting a parameterized auxiliary linear operator can be extremely useful for minimizing residual errors when used concurrently with the optimal homotopy analysis method, suggesting that this approach can prove useful for a number of nonlinear partial differential equations arising in physics and nonlinear mechanics.
Fedele, Renato; De Nicola, Sergio; Grecu, Dan; Visinescu, Anca; Shukla, Padma K.
2009-11-10
A review of the recent studies on the correspondence between a wide family of the generalized nonlinear Schroedinger equations and a wide family of the generalized Korteweg-de Vries equations is presented. It was constructed some years ago within the framework of a recently-developed approach based on the Madelung's fluid representation of the generalized nonlinear Schroedinger equation. The present analysis extends the former approach, developed for nonlinear Schroedinger equation with a nonlinear term proportional to a multiplicative operator, to the cases of derivative operators and the ones corresponding to cylindrical nonlinear Schroedinger equations.
Javan, N. Sepehri Homami, S. H. H.
2015-02-15
Self-guided nonlinear propagation of intense circularly-polarized electromagnetic waves in a hot electron-positron-ion magnetoplasma is studied. Using a relativistic fluid model, a nonlinear equation is derived, which describes the interaction of the electromagnetic wave with the plasma in the quasi-neutral approximation. Transverse Eigen modes, the nonlinear dispersion relation and the group velocity are obtained. Results show that the transverse profile in the case of magnetized plasma with cylindrical symmetry has a radially damping oscillatory form. Effect of applying external magnetic fields, existence of the electron-positron pairs, changing the amplitude of the electromagnetic wave, and its polarization on the nonlinear dispersion relation and Eigen modes are studied.
Shock-wave structure using nonlinear model Boltzmann equations.
NASA Technical Reports Server (NTRS)
Segal, B. M.; Ferziger, J. H.
1972-01-01
The structure of strong plane shock waves in a perfect monatomic gas was studied using four nonlinear models of the Boltzmann equation. The models involved the use of a simplified collision operator with velocity-independent collision frequency, in place of the complicated Boltzmann collision operator. The models employed were the BGK and ellipsoidal models developed by earlier authors, and the polynomial and trimodal gain function models developed during the work. An exact set of moment equations was derived for the density, velocity, temperature, viscous stress, and heat flux within the shock. This set was reduced to a pair of coupled nonlinear integral equations and solved using specially adapted numerical techniques. A new and simple Gauss-Seidel iteration was developed during the work and found to be as efficient as the best earlier iteration methods.
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.
Transformation matrices between non-linear and linear differential equations
NASA Technical Reports Server (NTRS)
Sartain, R. L.
1983-01-01
In the linearization of systems of non-linear differential equations, those systems which can be exactly transformed into the second order linear differential equation Y"-AY'-BY=0 where Y, Y', and Y" are n x 1 vectors and A and B are constant n x n matrices of real numbers were considered. The 2n x 2n matrix was used to transform the above matrix equation into the first order matrix equation X' = MX. Specially the matrix M and the conditions which will diagonalize or triangularize M were studied. Transformation matrices P and P sub -1 were used to accomplish this diagonalization or triangularization to return to the solution of the second order matrix differential equation system from the first order system.
Effects Of Relative Strength Of Dispersion On The Formation Of Nonlinear Waves In Dusty Plasmas
Asgari, H.; Muniandy, S. V.; Wong, C. S.; Yap, S. L.
2009-07-07
In this paper, we studied the effect of strength of dispersion on the formation of solitons and shock waves in un-magnetized dusty plasma using the reductive perturbative technique. Different relational forms of strength parameter epsilon were chosen such a way that it altered the stretching of space, x and time, t variables, thereby leading to different nonlinearities. First, we considered the form zeta = sq root(epsilon(x-v{sub 0}t)) and tau = sq root(epsilont), where v{sub 0} is the phase velocity, with 0
The dispersion equation of the induced Smith-Purcell instability
NASA Astrophysics Data System (ADS)
Klochkov, Dmitry N.; Artemyev, Alexander I.; Oganesyan, Koryun B.; Rostovtsev, Yuri V.; Scully, Marlan O.; Hu, Chin-Kun
2010-09-01
The simplest model of a magnetized infinitely thin electron beam is considered. For the grating, where the depth of grooves is a small parameter, the dispersion equation of the induced Smith-Purcell instability was obtained. It was found that the condition of the Thompson or the Raman regimes of excitation does not depend on beam current but depends on the height of the beam above the grating surface. The growth rate of instability in both cases is proportional to the square root of the electron beam current.
Phase space lattices and integrable nonlinear wave equations
NASA Astrophysics Data System (ADS)
Tracy, Eugene; Zobin, Nahum
2003-10-01
Nonlinear wave equations in fluids and plasmas that are integrable by Inverse Scattering Theory (IST), such as the Korteweg-deVries and nonlinear Schrodinger equations, are known to be infinite-dimensional Hamiltonian systems [1]. These are of interest physically because they predict new phenomena not present in linear wave theories, such as solitons and rogue waves. The IST method provides solutions of these equations in terms of a special class of functions called Riemann theta functions. The usual approach to the theory of theta functions tends to obscure the underlying phase space structure. A theory due to Mumford and Igusa [2], however shows that the theta functions arise naturally in the study of phase space lattices. We will describe this theory, as well as potential applications to nonlinear signal processing and the statistical theory of nonlinear waves. 1] , S. Novikov, S. V. Manakov, L. P. Pitaevskii and V. E. Zakharov, Theory of solitons: the inverse scattering method (Consultants Bureau, New York, 1984). 2] D. Mumford, Tata lectures on theta, Vols. I-III (Birkhauser); J. Igusa, Theta functions (Springer-Verlag, New York, 1972).
An adaptive grid algorithm for one-dimensional nonlinear equations
NASA Technical Reports Server (NTRS)
Gutierrez, William E.; Hills, Richard G.
1990-01-01
Richards' equation, which models the flow of liquid through unsaturated porous media, is highly nonlinear and difficult to solve. Step gradients in the field variables require the use of fine grids and small time step sizes. The numerical instabilities caused by the nonlinearities often require the use of iterative methods such as Picard or Newton interation. These difficulties result in large CPU requirements in solving Richards equation. With this in mind, adaptive and multigrid methods are investigated for use with nonlinear equations such as Richards' equation. Attention is focused on one-dimensional transient problems. To investigate the use of multigrid and adaptive grid methods, a series of problems are studied. First, a multigrid program is developed and used to solve an ordinary differential equation, demonstrating the efficiency with which low and high frequency errors are smoothed out. The multigrid algorithm and an adaptive grid algorithm is used to solve one-dimensional transient partial differential equations, such as the diffusive and convective-diffusion equations. The performance of these programs are compared to that of the Gauss-Seidel and tridiagonal methods. The adaptive and multigrid schemes outperformed the Gauss-Seidel algorithm, but were not as fast as the tridiagonal method. The adaptive grid scheme solved the problems slightly faster than the multigrid method. To solve nonlinear problems, Picard iterations are introduced into the adaptive grid and tridiagonal methods. Burgers' equation is used as a test problem for the two algorithms. Both methods obtain solutions of comparable accuracy for similar time increments. For the Burgers' equation, the adaptive grid method finds the solution approximately three times faster than the tridiagonal method. Finally, both schemes are used to solve the water content formulation of the Richards' equation. For this problem, the adaptive grid method obtains a more accurate solution in fewer work units and
Travelling Waves for the Nonlinear Schrödinger Equation with General Nonlinearity in Dimension Two
NASA Astrophysics Data System (ADS)
Chiron, David; Scheid, Claire
2016-02-01
We investigate numerically the two-dimensional travelling waves of the nonlinear Schrödinger equation for a general nonlinearity and with nonzero condition at infinity. In particular, we are interested in the energy-momentum diagrams. We propose a numerical strategy based on the variational structure of the equation. The key point is to characterize the saddle points of the action as minimizers of another functional that allows us to use a gradient flow. We combine this approach with a continuation method in speed in order to obtain the full range of velocities. Through various examples, we show that even though the nonlinearity has the same behaviour as the well-known Gross-Pitaevskii nonlinearity, the qualitative properties of the travelling waves may be extremely different. For instance, we observe cusps, a modified KP-I asymptotic in the transonic limit, various multiplicity results and "one-dimensional spreading" phenomena.
Collapse for the higher-order nonlinear Schrödinger equation
Achilleos, V.; Diamantidis, S.; Frantzeskakis, D. J.; Horikis, T. P.; Karachalios, N. I.; Kevrekidis, P. G.
2016-02-01
We examine conditions for finite-time collapse of the solutions of the higher-order nonlinear Schr odinger (NLS) equation incorporating third-order dispersion, self-steepening, linear and nonlinear gain and loss, and Raman scattering; this is a system that appears in many physical contexts as a more realistic generalization of the integrable NLS. By using energy arguments, it is found that the collapse dynamics is chiefly controlled by the linear/nonlinear gain/loss strengths. We identify a critical value of the linear gain, separating the possible decay of solutions to the trivial zero-state, from collapse. The numerical simulations, performed for a wide class of initial data,more » are found to be in very good agreement with the analytical results, and reveal long-time stability properties of localized solutions. The role of the higher-order effects to the transient dynamics is also revealed in these simulations.« less
Chirped Peregrine solitons in a class of cubic-quintic nonlinear Schrödinger equations
NASA Astrophysics Data System (ADS)
Chen, Shihua; Baronio, Fabio; Soto-Crespo, Jose M.; Liu, Yi; Grelu, Philippe
2016-06-01
We shed light on the fundamental form of the Peregrine soliton as well as on its frequency chirping property by virtue of a pertinent cubic-quintic nonlinear Schrödinger equation. An exact generic Peregrine soliton solution is obtained via a simple gauge transformation, which unifies the recently-most-studied fundamental rogue-wave species. We discover that this type of Peregrine soliton, viable for both the focusing and defocusing Kerr nonlinearities, could exhibit an extra doubly localized chirp while keeping the characteristic intensity features of the original Peregrine soliton, hence the term chirped Peregrine soliton. The existence of chirped Peregrine solitons in a self-defocusing nonlinear medium may be attributed to the presence of self-steepening effect when the latter is not balanced out by the third-order dispersion. We numerically confirm the robustness of such chirped Peregrine solitons in spite of the onset of modulation instability.
Kenmogne, Fabien; Yemélé, David
2013-10-01
The modulational instability (MI) phenomenon in the nonlinear Schrödinger equation (NLSE) extended by two different nonlinear dispersion terms and the gradient term is investigated. We find that the possibility of instability of plane waves depends on the sign of the nonlinear dispersion parameters with regard to the linear dispersion coefficient. In contrast to the basic NLSE, the system may exhibit instability in the defocusing media for amplitude exceeding a critical value depending on the magnitude of the nonlinear dispersion. An additional feature, namely the higher order or the infinite gain band, absent in the NLSE case, may appear and in which MI induces the birth of the nonlinear localized wave (NLW) of different carrier wave numbers. The result of the qualitative investigations of the system's dynamics indicates the existence of the NLW, such as peak, bright, dark, and compact dark solitary waves which can be well predicted by the MI criteria. In addition the nonlinear dispersion induces the existence of a pair of bright-dark solitary waves which is usually exhibited by the coupled NLSEs only, and the pairs of peak-dark and compact dark-bright solitary waves. PMID:24229297
Dispersion and nonlinear effects in OFDM-RoF system
NASA Astrophysics Data System (ADS)
Alhasson, Bader H.; Bloul, Albe M.; Matin, M.
2010-08-01
The radio-over-fiber (RoF) network has been a proven technology to be the best candidate for the wireless-access technology, and the orthogonal frequency division multiplexing (OFDM) technique has been established as the core technology in the physical layer of next generation wireless communication system, as a result OFDM-RoF has drawn attentions worldwide and raised many new research topics recently. At the present time, the trend of information industry is towards mobile, wireless, digital and broadband. The next generation network (NGN) has motivated researchers to study higher-speed wider-band multimedia communication to transmit (voice, data, and all sorts of media such as video) at a higher speed. The NGN would offer services that would necessitate broadband networks with bandwidth higher than 2Mbit/s per radio channel. Many new services emerged, such as Internet Protocol TV (IPTV), High Definition TV (HDTV), mobile multimedia and video stream media. Both speed and capacity have been the key objectives in transmission. In the meantime, the demand for transmission bandwidth increased at a very quick pace. The coming of 4G and 5G era will provide faster data transmission and higher bit rate and bandwidth. Taking advantages of both optical communication and wireless communication, OFDM Radio over Fiber (OFDM-RoF) system is characterized by its high speed, large capacity and high spectral efficiency. However, up to the present there are some problems to be solved, such as dispersion and nonlinearity effects. In this paper we will study the dispersion and nonlinearity effects and their elimination in OFDM-radio-over-fiber system.
Golush, W.G.
1994-12-31
Nonlinear equations are expressed using a new OMNI statement FORM NLE. This allows OMNI Constructs, Classes, Tables, and New Variables to be used in nonlinear equations. The interface passes the nonlinear equations and symbolic derivatives to a general nonlinear solver. After optimization, the row and column activities of the solution are written to an OMNI Standard Solution File. Reports are written from this file using the OMNI FORM LINE report writer. The interface will be illustrated with an example of a nonlinear model written in OMNI and solved using the MINOS nonlinear solver.
Multi-soliton rational solutions for some nonlinear evolution equations
NASA Astrophysics Data System (ADS)
Osman, Mohamed S.
2016-01-01
The Korteweg-de Vries equation (KdV) and the (2+ 1)-dimensional Nizhnik-Novikov-Veselov system (NNV) are presented. Multi-soliton rational solutions of these equations are obtained via the generalized unified method. The analysis emphasizes the power of this method and its capability of handling completely (or partially) integrable equations. Compared with Hirota's method and the inverse scattering method, the proposed method gives more general exact multi-wave solutions without much additional effort. The results show that, by virtue of symbolic computation, the generalized unified method may provide us with a straightforward and effective mathematical tool for seeking multi-soliton rational solutions for solving many nonlinear evolution equations arising in different branches of sciences.
Numerical solution of control problems governed by nonlinear differential equations
Heinkenschloss, M.
1994-12-31
In this presentation the author investigates an iterative method for the solution of optimal control problems. These problems are formulated as constrained optimization problems with constraints arising from the state equation and in the form of bound constraints on the control. The method for the solution of these problems uses the special structure of the problem arising from the bound constraint and the state equation. It is derived from SQP methods and projected Newton methods and combines the advantages of both methods. The bound constraint is satisfied by all iterates using a projection, the nonlinear state equation is satisfied in the limit. Only a linearized state equation has to be solved in every iteration. The solution of the linearized problems are done using multilevel methods and GMRES.
Solving nonlinear evolution equation system using two different methods
NASA Astrophysics Data System (ADS)
Kaplan, Melike; Bekir, Ahmet; Ozer, Mehmet N.
2015-12-01
This paper deals with constructing more general exact solutions of the coupled Higgs equation by using the (G0/G, 1/G)-expansion and (1/G0)-expansion methods. The obtained solutions are expressed by three types of functions: hyperbolic, trigonometric and rational functions with free parameters. It has been shown that the suggested methods are productive and will be used to solve nonlinear partial differential equations in applied mathematics and engineering. Throughout the paper, all the calculations are made with the aid of the Maple software.
Quadratic nonlinear Klein-Gordon equation in one dimension
NASA Astrophysics Data System (ADS)
Hayashi, Nakao; Naumkin, Pavel I.
2012-10-01
We study the initial value problem for the quadratic nonlinear Klein-Gordon equation vtt + v - vxx = λv2, t ∈ R, x ∈ R, with initial conditions v(0, x) = v0(x), vt(0, x) = v1(x), x ∈ R, where v0 and v1 are real-valued functions, λ ∈ R. Using the method of normal forms of Shatah ["Normal forms and quadratic nonlinear Klein-Gordon equations," Commun. Pure Appl. Math. 38, 685-696 (1985)], we obtain a sharp asymptotic behavior of small solutions without the condition of a compact support on the initial data, which was assumed in the previous work of J.-M. Delort ["Existence globale et comportement asymptotique pour l'équation de Klein-Gordon quasi-linéaire á données petites en dimension 1," Ann. Sci. Ec. Normale Super. 34(4), 1-61 (2001)].
Unitary qubit extremely parallelized algorithms for coupled nonlinear Schrodinger equations
NASA Astrophysics Data System (ADS)
Oganesov, Armen; Flint, Chris; Vahala, George; Vahala, Linda; Yepez, Jeffrey; Soe, Min
2015-11-01
The nonlinear Schrodinger equation (NLS) is a ubiquitous equation occurring in plasma physics, nonlinear optics and in Bose Einstein condensates. Viewed from the BEC standpoint of phase transitions, the wave function is the order parameter and topological defects in that manifold are simply the vortices, which for a scalar NLS have quantized circulation. In multi-species NLS the topological nature of the vortices are radically different with some classes of vortices no longer having quantized circulation as in classical turbulence. Moreover, some of the vortex equivalence classes need no longer be Abelian. This strongly effects the permitted vortex reconnections. The effect of these structures on the spectral properties of the ensuing turbulence will be investigated. Our 3D algorithm is based on a novel unitary qubit lattice scheme that is ideally parallelized - tested up to 780 000 cores on Mira. This scheme is mesoscopic (like lattice Boltzmann), but fully unitary (unlike LB). Supported by NSF, DoD.
Parallel iterative methods for sparse linear and nonlinear equations
NASA Technical Reports Server (NTRS)
Saad, Youcef
1989-01-01
As three-dimensional models are gaining importance, iterative methods will become almost mandatory. Among these, preconditioned Krylov subspace methods have been viewed as the most efficient and reliable, when solving linear as well as nonlinear systems of equations. There has been several different approaches taken to adapt iterative methods for supercomputers. Some of these approaches are discussed and the methods that deal more specifically with general unstructured sparse matrices, such as those arising from finite element methods, are emphasized.
Fast neural solution of a nonlinear wave equation
NASA Technical Reports Server (NTRS)
Toomarian, Nikzad; Barhen, Jacob
1992-01-01
A neural algorithm for rapidly simulating a certain class of nonlinear wave phenomena using analog VLSI neural hardware is presented and applied to the Korteweg-de Vries partial differential equation. The corresponding neural architecture is obtained from a pseudospectral representation of the spatial dependence, along with a leap-frog scheme for the temporal evolution. Numerical simulations demonstrated the robustness of the proposed approach.
Approximate solutions for non-linear iterative fractional differential equations
NASA Astrophysics Data System (ADS)
Damag, Faten H.; Kiliçman, Adem; Ibrahim, Rabha W.
2016-06-01
This paper establishes approximate solution for non-linear iterative fractional differential equations: d/γv (s ) d sγ =ℵ (s ,v ,v (v )), where γ ∈ (0, 1], s ∈ I := [0, 1]. Our method is based on some convergence tools for analytic solution in a connected region. We show that the suggested solution is unique and convergent by some well known geometric functions.
Inhomogeneous critical nonlinear Schroedinger equations with a harmonic potential
Cao Daomin; Han Pigong
2010-04-15
In this paper, we study the Cauchy problem of the inhomogeneous nonlinear Schroedinger equation with a harmonic potential: i{partial_derivative}{sub t}u=-div(f(x){nabla}u)+|x|{sup 2}u-k(x)|u|{sup 4/N}u, x is an element of R{sup N}, N{>=}1, which models the remarkable Bose-Einstein condensation. We discuss the existence and nonexistence results and investigate the limiting profile of blow-up solutions with critical mass.
Nonlinear Generalized Hydrodynamic Wave Equations in Strongly Coupled Dusty Plasmas
Veeresha, B. M.; Sen, A.; Kaw, P. K.
2008-09-07
A set of nonlinear equations for the study of low frequency waves in a strongly coupled dusty plasma medium is derived using the phenomenological generalized hydrodynamic (GH) model and is used to study the modulational stability of dust acoustic waves to parallel perturbations. Dust compressibility contributions arising from strong Coulomb coupling effects are found to introduce significant modifications in the threshold and range of the instability domain.
NASA Astrophysics Data System (ADS)
Tamizhmani, K. M.; Krishnakumar, K.; Leach, P. G. L.
2015-11-01
We examine the reductions of the order of certain third- and second-order nonlinear equations with arbitrary nonlinearity through their symmetries and some appropriate transformations. We use the folding transformation which enables one to change from a nonlinearity with an arbitrary exponent to a nonlinearity with a specific numerical exponent.
Improved algorithm for solving nonlinear parabolized stability equations
NASA Astrophysics Data System (ADS)
Zhao, Lei; Zhang, Cun-bo; Liu, Jian-xin; Luo, Ji-sheng
2016-08-01
Due to its high computational efficiency and ability to consider nonparallel and nonlinear effects, nonlinear parabolized stability equations (NPSE) approach has been widely used to study the stability and transition mechanisms. However, it often diverges in hypersonic boundary layers when the amplitude of disturbance reaches a certain level. In this study, an improved algorithm for solving NPSE is developed. In this algorithm, the mean flow distortion is included into the linear operator instead of into the nonlinear forcing terms in NPSE. An under-relaxation factor for computing the nonlinear terms is introduced during the iteration process to guarantee the robustness of the algorithm. Two case studies, the nonlinear development of stationary crossflow vortices and the fundamental resonance of the second mode disturbance in hypersonic boundary layers, are presented to validate the proposed algorithm for NPSE. Results from direct numerical simulation (DNS) are regarded as the baseline for comparison. Good agreement can be found between the proposed algorithm and DNS, which indicates the great potential of the proposed method on studying the crossflow and streamwise instability in hypersonic boundary layers. Project supported by the National Natural Science Foundation of China (Grant Nos. 11332007 and 11402167).
NASA Astrophysics Data System (ADS)
Xie, Xi-Yang; Tian, Bo; Wang, Yu-Feng; Sun, Ya; Jiang, Yan
2015-11-01
In this paper, we investigate a generalized nonautonomous nonlinear equation which describes the ultrashort optical pulse propagating in a nonlinear inhomogeneous fiber. By virtue of the generalized Darboux transformation, the first- and second-order rogue-wave solutions for the generalized nonautonomous nonlinear equation are obtained, under some variable-coefficient constraints. Properties of the first- and second-order rogue waves are graphically presented and analyzed: When the coefficients are all chosen as the constants, we can observe the some functions, the shapes of wave crests and troughs for the first- and second-order rogue waves change. Oscillating behaviors of the first- and second-order rogue waves are observed when the coefficients are the trigonometric functions.
Marhic, M E; Kagi, N; Chiang, T K; Kazovsky, L G
1995-04-15
We show that in principle it is possible to cancel third-order nonlinear effects in optical fiber links. The necessary conditions exist in two-segment links, with dispersion compensation, phase conjugation, and amplification between the two, as well as opposite chromatic dispersion coefficients in the segments. The cancellation is independent of loss, modulation format, and modulation frequency. PMID:19859355
NASA Astrophysics Data System (ADS)
Zakeri, Gholam-Ali; Yomba, Emmanuel
2015-06-01
A generalized (2+1)-dimensional coupled cubic-quintic Ginzburg-Landau equation with higher-order nonlinearities is fully investigated for modulational instability regions. We obtained the constraints that allow the modulational instability (MI) procedure to transform the system under consideration into an analysis of the roots of a polynomial equation of the fourth degree. Because of the complexity of the dispersion relation and its dependence on many parameters, we study numerous examples that are presented graphically. A numerical simulation based on a split-step Fourier method is implemented on the above equation. In addition to the general case, we have considered some special cases that allow us to investigate the behavior of MI in different regions.
Zakeri, Gholam-Ali; Yomba, Emmanuel
2015-06-01
A generalized (2+1)-dimensional coupled cubic-quintic Ginzburg-Landau equation with higher-order nonlinearities is fully investigated for modulational instability regions. We obtained the constraints that allow the modulational instability (MI) procedure to transform the system under consideration into an analysis of the roots of a polynomial equation of the fourth degree. Because of the complexity of the dispersion relation and its dependence on many parameters, we study numerous examples that are presented graphically. A numerical simulation based on a split-step Fourier method is implemented on the above equation. In addition to the general case, we have considered some special cases that allow us to investigate the behavior of MI in different regions. PMID:26172769
Compensating for dispersion and the nonlinear Kerr effect without phase conjugation.
Paré, C; Villeneuve, A; Bélanger, P A; Doran, N J
1996-04-01
We propose the use of a dispersive medium with a negative nonlinear refractive-index coefficient as a way to compensate for the dispersion and the nonlinear effects resulting from pulse propagation in an optical fiber. The undoing of pulse interaction might allow for increased bit rates. PMID:19865438
Nonlinear electromagnetic gyrokinetic equations for rotating axisymmetric plasmas
Artun, M.; Tang, W.M.
1994-03-01
The influence of sheared equilibrium flows on the confinement properties of tokamak plasmas is a topic of much current interest. A proper theoretical foundation for the systematic kinetic analysis of this important problem has been provided here by presented the derivation of a set of nonlinear electromagnetic gyrokinetic equations applicable to low frequency microinstabilities in a rotating axisymmetric plasma. The subsonic rotation velocity considered is in the direction of symmetry with the angular rotation frequency being a function of the equilibrium magnetic flux surface. In accordance with experimental observations, the rotation profile is chosen to scale with the ion temperature. The results obtained represent the shear flow generalization of the earlier analysis by Frieman and Chen where such flows were not taken into account. In order to make it readily applicable to gyrokinetic particle simulations, this set of equations is cast in a phase-space-conserving continuity equation form.
Equations for Nonlinear MHD Convection in Shearless Magnetic Systems
Pastukhov, V.P.
2005-07-15
A closed set of reduced dynamic equations is derived that describe nonlinear low-frequency flute MHD convection and resulting nondiffusive transport processes in weakly dissipative plasmas with closed or open magnetic field lines. The equations obtained make it possible to self-consistently simulate transport processes and the establishment of the self-consistent plasma temperature and density profiles for a large class of axisymmetric nonparaxial shearless magnetic devices: levitated dipole configurations, mirror systems, compact tori, etc. Reduced equations that are suitable for modeling the long-term evolution of the plasma on time scales comparable to the plasma lifetime are derived by the method of the adiabatic separation of fast and slow motions.
Multipulses of Nonlinearly Coupled Schrödinger Equations
NASA Astrophysics Data System (ADS)
Yew, Alice C.
2001-06-01
The capacity of coupled nonlinear Schrödinger (NLS) equations to support multipulse solutions (multibump solitary-waves) is investigated. A detailed analysis is undertaken for a system of quadratically coupled equations that describe the phenomena of second harmonic generation and parametric wave interaction in non-centrosymmetric optical materials. Utilising the framework of homoclinic bifurcation theory, and employing a Lyapunov-Schmidt reduction method developed by Hale, Lin, and Sandstede, a novel mechanism for the generation of multipulses is identified, which arises from a resonant semi-simple eigenvalue configuration of the linearised steady-state equations. Conditions for the existence of multipulses, as well as a description of their geometry, are derived from the analysis.
Solovchuk, Maxim; Sheu, Tony W H; Thiriet, Marc
2013-11-01
This study investigates the influence of blood flow on temperature distribution during high-intensity focused ultrasound (HIFU) ablation of liver tumors. A three-dimensional acoustic-thermal-hydrodynamic coupling model is developed to compute the temperature field in the hepatic cancerous region. The model is based on the nonlinear Westervelt equation, bioheat equations for the perfused tissue and blood flow domains. The nonlinear Navier-Stokes equations are employed to describe the flow in large blood vessels. The effect of acoustic streaming is also taken into account in the present HIFU simulation study. A simulation of the Westervelt equation requires a prohibitively large amount of computer resources. Therefore a sixth-order accurate acoustic scheme in three-point stencil was developed for effectively solving the nonlinear wave equation. Results show that focused ultrasound beam with the peak intensity 2470 W/cm(2) can induce acoustic streaming velocities up to 75 cm/s in the vessel with a diameter of 3 mm. The predicted temperature difference for the cases considered with and without acoustic streaming effect is 13.5 °C or 81% on the blood vessel wall for the vein. Tumor necrosis was studied in a region close to major vessels. The theoretical feasibility to safely necrotize the tumors close to major hepatic arteries and veins was shown. PMID:24180802
The method of patches for solving stiff nonlinear differential equations
NASA Astrophysics Data System (ADS)
Brydon, David Van George, Jr.
1998-12-01
This dissertation describes a new method for solving very stiff sets of ordinary differential equations. The basic idea is to replace the original nonlinear equations with a set of equally stiff equations that are piecewise linear, and therefore can be solved exactly. We demonstrate the value of the method on small systems of equations for which some other methods are inefficient or produce spurious solutions, estimate error bounds, and discuss extensions of the method to larger systems of equations and to partial differential equations. Putzer's method is developed in a novel way for efficient and accurate solution of dx/dt = Ax+b. The physical problem of interest is spatial pattern formation in open reaction-diffusion chemical systems, as studied in the experiments of Kyoung Lee, Harry Swinney, et al. I develop a new experiment model that agrees reasonably well with experimental results. I solve the model, applying the new method to the two-variable Gaspar- Showalter chemical kinetics in two space dimensions. Because of time and computer limitations, only preliminary pattern-formation results are achieved and reported.
All-fiber nonlinearity- and dispersion-managed dissipative soliton nanotube mode-locked laser
Zhang, Z.; Popa, D. Wittwer, V. J.; Milana, S.; Hasan, T.; Jiang, Z.; Ferrari, A. C.; Ilday, F. Ö.
2015-12-14
We report dissipative soliton generation from an Yb-doped all-fiber nonlinearity- and dispersion-managed nanotube mode-locked laser. A simple all-fiber ring cavity exploits a photonic crystal fiber for both nonlinearity enhancement and dispersion compensation. The laser generates stable dissipative solitons with large linear chirp in the net normal dispersion regime. Pulses that are 8.7 ps long are externally compressed to 118 fs, outperforming current nanotube-based Yb-doped fiber laser designs.
Wei, Haiqing; Plant, David V
2005-09-15
A method of packaging dispersion-compensating fibers (DCFs) is discussed that achieves optimal nonlinearity compensation and a good signal-to-noise ratio simultaneously. An optimally packaged dispersion-compensating module (DCM) may consist of portions of DCFs with higher and lower loss coefficients. Such optimized DCMs may be paired with transmission fibers to form scaled translation-symmetric lines that could effectively compensate for signal distortions due to dispersion and nonlinearity, with or without optical phase conjugation. PMID:16196322
The truncation model of the derivative nonlinear Schroedinger equation
Sanchez-Arriaga, G.; Hada, T.; Nariyuki, Y.
2009-04-15
The derivative nonlinear Schroedinger (DNLS) equation is explored using a truncation model with three resonant traveling waves. In the conservative case, the system derives from a time-independent Hamiltonian function with only one degree of freedom and the solutions can be written using elliptic functions. In spite of its low dimensional order, the truncation model preserves some features from the DNLS equation. In particular, the modulational instability criterion fits with the existence of two hyperbolic fixed points joined by a heteroclinic orbit that forces the exchange of energy between the three waves. On the other hand, numerical integrations of the DNLS equation show that the truncation model fails when wave energy is increased or left-hand polarized modulational unstable modes are present. When dissipative and growth terms are added the system exhibits a very complex dynamics including appearance of several attractors, period doubling bifurcations leading to chaos as well as other nonlinear phenomenon. In this case, the validity of the truncation model depends on the strength of the dissipation and the kind of attractor investigated.
Theoretical and numerical studies of nonlinear shell equations
NASA Astrophysics Data System (ADS)
Hermann, M.; Kaiser, D.; Schröder, M.
1999-07-01
We study the solution field M of a parameter dependent nonlinear two-point boundary value problem presented by Troger and Steindl [H. Troger, A. Steindl, Nonlinear Stability and Bifurcation Theory, Springer, Wien, New York, 1991]. This problem models the buckling of a thin-walled spherical shell under a uniform external static pressure. The boundary value problem is formulated as an abstract operator equation T( x, λ)=0 in appropriate Banach spaces. By exploiting the equivariance of T, we obtain detailed informations about the structure of M. These theoretical results are used to compute efficiently interesting parts of M with numerical standard techniques. Bifurcation diagrams, a stability diagram and pictures of deformed shells are presented.
NONLINEAR SIMULATION OF TURBULENT FIELD LINES: DISPERSAL STATISTICS
Ragot, B. R.
2010-11-10
A new method for the full nonlinear computation of sets of turbulent field lines is introduced that extends the sums of random numbers distribution method previously applied to the computation of individual field lines. With a multiscale variation of the phases consistent with in situ observations of intermittent solar wind (SW) turbulence, the new method allows inclusion of the equivalent of more than four decades of turbulent scales with a fully three-dimensional distribution of wavevectors. As a first application, pairs of magnetic field lines are computed in independent realizations of the turbulence, for spectra typical of the quiet slow SW near 1 AU. The statistics of field-line dispersal are then studied from the simulated pairs of magnetic field lines and compared to earlier theoretical predictions. It appears that while the earlier theoretical picture remains relatively accurate as long as the mean variation of separation logarithm {Lambda} is less than one, the qualitative picture is quickly altered as {Lambda} grows past one.
Sun Zhiyuan; Yu Xin; Liu Ying; Gao Yitian
2012-12-15
We investigate the dynamics of the bound vector solitons (BVSs) for the coupled nonlinear Schroedinger equations with the nonhomogenously stochastic perturbations added on their dispersion terms. Soliton switching (besides soliton breakup) can be observed between the two components of the BVSs. Rate of the maximum switched energy (absolute values) within the fixed propagation distance (about 10 periods of the BVSs) enhances in the sense of statistics when the amplitudes of stochastic perturbations increase. Additionally, it is revealed that the BVSs with enhanced coherence are more robust against the perturbations with nonhomogenous stochasticity. Diagram describing the approximate borders of the splitting and non-splitting areas is also given. Our results might be helpful in dynamics of the BVSs with stochastic noises in nonlinear optical fibers or with stochastic quantum fluctuations in Bose-Einstein condensates.
Pseudorecurrence and chaos of cubic-quintic nonlinear Schroedinger equation
Zhou, C.; Lai, C.H.
1996-12-01
Recurrence, pseudorecurrence, and chaotic solutions for a continuum Hamiltonian system in which there exist spatial patterns of solitary wave structures are investigated using the nonlinear Schrodinger equation (NSE) with cubic and quintic terms. The theoretical analyses indicate that there may exist Birkhoff`s recurrence for the arbitrary parameter values. The numerical experiments show that there may be Fermi-Pasta-Ulam (FPU) recurrence, pseudorecurrence, and chaos when different initial conditions are chosen. The fact that the system energy is effectively shared by finite Fourier modes suggests that it may be possible to describe the continuum system in terms of some effective degrees of freedom.
Vortex Solutions of the Defocusing Discrete Nonlinear Schroedinger Equation
Cuevas, J.; Kevrekidis, P. G.; Law, K. J. H.
2009-09-09
We consider the existence, stability and dynamical evolution of dark vortex states in the two-dimensional defocusing DNLS equation, a model of interest both to atomic physics and to nonlinear optics. Our considerations are chiefly based on initializing such vortex configurations at the anti-continuum limit of zero coupling between adjacent sites, and continuing them to finite values of the coupling. Discrete defocusing vortices become unstable past a critical coupling strength and, subsequently feature a cascade of alternating stabilization-destabilization windows for any finite lattice.
Superposition of elliptic functions as solutions for a large number of nonlinear equations
Khare, Avinash; Saxena, Avadh
2014-03-15
For a large number of nonlinear equations, both discrete and continuum, we demonstrate a kind of linear superposition. We show that whenever a nonlinear equation admits solutions in terms of both Jacobi elliptic functions cn(x, m) and dn(x, m) with modulus m, then it also admits solutions in terms of their sum as well as difference. We have checked this in the case of several nonlinear equations such as the nonlinear Schrödinger equation, MKdV, a mixed KdV-MKdV system, a mixed quadratic-cubic nonlinear Schrödinger equation, the Ablowitz-Ladik equation, the saturable nonlinear Schrödinger equation, λϕ{sup 4}, the discrete MKdV as well as for several coupled field equations. Further, for a large number of nonlinear equations, we show that whenever a nonlinear equation admits a periodic solution in terms of dn{sup 2}(x, m), it also admits solutions in terms of dn {sup 2}(x,m)±√(m) cn (x,m) dn (x,m), even though cn(x, m)dn(x, m) is not a solution of these nonlinear equations. Finally, we also obtain superposed solutions of various forms for several coupled nonlinear equations.
Stochastic Computational Approach for Complex Nonlinear Ordinary Differential Equations
NASA Astrophysics Data System (ADS)
Junaid, Ali Khan; Muhammad, Asif Zahoor Raja; Ijaz Mansoor, Qureshi
2011-02-01
We present an evolutionary computational approach for the solution of nonlinear ordinary differential equations (NLODEs). The mathematical modeling is performed by a feed-forward artificial neural network that defines an unsupervised error. The training of these networks is achieved by a hybrid intelligent algorithm, a combination of global search with genetic algorithm and local search by pattern search technique. The applicability of this approach ranges from single order NLODEs, to systems of coupled differential equations. We illustrate the method by solving a variety of model problems and present comparisons with solutions obtained by exact methods and classical numerical methods. The solution is provided on a continuous finite time interval unlike the other numerical techniques with comparable accuracy. With the advent of neuroprocessors and digital signal processors the method becomes particularly interesting due to the expected essential gains in the execution speed.
Difference equation state approximations for nonlinear hereditary control problems
NASA Technical Reports Server (NTRS)
Rosen, I. G.
1984-01-01
Discrete approximation schemes for the solution of nonlinear hereditary control problems are constructed. The methods involve approximation by a sequence of optimal control problems in which the original infinite dimensional state equation has been approximated by a finite dimensional discrete difference equation. Convergence of the state approximations is argued using linear semigroup theory and is then used to demonstrate that solutions to the approximating optimal control problems in some sense approximate solutions to the original control problem. Two schemes, one based upon piecewise constant approximation, and the other involving spline functions are discussed. Numerical results are presented, analyzed and used to compare the schemes to other available approximation methods for the solution of hereditary control problems. Previously announced in STAR as N83-33589
Difference equation state approximations for nonlinear hereditary control problems
NASA Technical Reports Server (NTRS)
Rosen, I. G.
1982-01-01
Discrete approximation schemes for the solution of nonlinear hereditary control problems are constructed. The methods involve approximation by a sequence of optimal control problems in which the original infinite dimensional state equation has been approximated by a finite dimensional discrete difference equation. Convergence of the state approximations is argued using linear semigroup theory and is then used to demonstrate that solutions to the approximating optimal control problems in some sense approximate solutions to the original control problem. Two schemes, one based upon piecewise constant approximation, and the other involving spline functions are discussed. Numerical results are presented, analyzed and used to compare the schemes to other available approximation methods for the solution of hereditary control problems.
New variable separation solutions for the generalized nonlinear diffusion equations
NASA Astrophysics Data System (ADS)
Fei-Yu, Ji; Shun-Li, Zhang
2016-03-01
The functionally generalized variable separation of the generalized nonlinear diffusion equations ut = A(u,ux)uxx + B(u,ux) is studied by using the conditional Lie-Bäcklund symmetry method. The variant forms of the considered equations, which admit the corresponding conditional Lie-Bäcklund symmetries, are characterized. To construct functionally generalized separable solutions, several concrete examples defined on the exponential and trigonometric invariant subspaces are provided. Project supported by the National Natural Science Foundation of China (Grant Nos. 11371293, 11401458, and 11501438), the National Natural Science Foundation of China, Tian Yuan Special Foundation (Grant No. 11426169), and the Natural Science Basic Research Plan in Shaanxi Province of China (Grant No. 2015JQ1014).
Nonlinear stability of oscillatory pulses in the parametric nonlinear Schrödinger equation
NASA Astrophysics Data System (ADS)
Chang, Paul A. C.; Promislow, Keith
2007-03-01
We extend the renormalization group method, developed for the study of pulse interaction in damped wave equations, to the study of oscillatory motion of supercritical pulses in the parametrically forced nonlinear Schrödinger equation (PNLS). We construct a global manifold which asymptotically attracts the flow into an {\\cal O}(r^4) neighbourhood in the H1 norm, where r is the amplitude of the internal oscillations. The oscillatory and translational dynamics of the pulses are rigorously recovered as a finite-dimensional flow on the manifold. The normal form for the projected dynamics of the oscillatory pulse shows that it is created in a supercritical Poincaré-Hopf bifurcation.
A Procedure to Construct Exact Solutions of Nonlinear Fractional Differential Equations
Güner, Özkan; Cevikel, Adem C.
2014-01-01
We use the fractional transformation to convert the nonlinear partial fractional differential equations with the nonlinear ordinary differential equations. The Exp-function method is extended to solve fractional partial differential equations in the sense of the modified Riemann-Liouville derivative. We apply the Exp-function method to the time fractional Sharma-Tasso-Olver equation, the space fractional Burgers equation, and the time fractional fmKdV equation. As a result, we obtain some new exact solutions. PMID:24737972
Parallel numerical integration of Maxwell's full-vector equations in nonlinear focusing media
NASA Astrophysics Data System (ADS)
Bennett, Paul Murray
Maxwell's equations governing the evolution of ultrashort intense coherent pulses of light in a nonlinear focusing dielectric are presented. A discretization of this model using Kane Yee's grid is presented. Initial and boundary conditions are derived, and a serial finite difference algorithm using Yee's grid with the initial and boundary conditions is given. A parallelization of the serial algorithm to more aptly handle the large computational size is performed, and speedup and efficiency results of the parallel program are presented. The parallel code is first used to study the effect of the focusing nonlinearity upon dispersionless pulse propagation. Indications are given of the development of shocks on the optical carrier wave and upon the pulse envelope. The parallel code is then used to study the effect of varying the focusing of the light by varying the intensity as a way to compensate linear dispersion. Blow-up of the pulse in finite propagation distance is demonstrated, and the dependence of the blow-up position upon the intensity of the light is presented. Optical saturation is considered to counter blow-up of intense pulses. Finally, the parallel code is used to study the evolution of intense ultrashort optical pulses in a model featuring nonlinear dispersion, focusing, and optical saturation.
Nonlinear periodic waves solutions of the nonlinear self-dual network equations
Laptev, Denis V. Bogdan, Mikhail M.
2014-04-15
The new classes of periodic solutions of nonlinear self-dual network equations describing the breather and soliton lattices, expressed in terms of the Jacobi elliptic functions have been obtained. The dependences of the frequencies on energy have been found. Numerical simulations of soliton lattice demonstrate their stability in the ideal lattice and the breather lattice instability in the dissipative lattice. However, the lifetime of such structures in the dissipative lattice can be extended through the application of ac driving terms.
Charged anisotropic matter with linear or nonlinear equation of state
Varela, Victor; Rahaman, Farook; Ray, Saibal; Chakraborty, Koushik; Kalam, Mehedi
2010-08-15
Ivanov pointed out substantial analytical difficulties associated with self-gravitating, static, isotropic fluid spheres when pressure explicitly depends on matter density. Simplifications achieved with the introduction of electric charge were noticed as well. We deal with self-gravitating, charged, anisotropic fluids and get even more flexibility in solving the Einstein-Maxwell equations. In order to discuss analytical solutions we extend Krori and Barua's method to include pressure anisotropy and linear or nonlinear equations of state. The field equations are reduced to a system of three algebraic equations for the anisotropic pressures as well as matter and electrostatic energy densities. Attention is paid to compact sources characterized by positive matter density and positive radial pressure. Arising solutions satisfy the energy conditions of general relativity. Spheres with vanishing net charge contain fluid elements with unbounded proper charge density located at the fluid-vacuum interface. Notably the electric force acting on these fluid elements is finite, although the acting electric field is zero. Net charges can be huge (10{sup 19}C) and maximum electric field intensities are very large (10{sup 23}-10{sup 24} statvolt/cm) even in the case of zero net charge. Inward-directed fluid forces caused by pressure anisotropy may allow equilibrium configurations with larger net charges and electric field intensities than those found in studies of charged isotropic fluids. Links of these results with charged strange quark stars as well as models of dark matter including massive charged particles are highlighted. The van der Waals equation of state leading to matter densities constrained by cubic polynomial equations is briefly considered. The fundamental question of stability is left open.
NASA Astrophysics Data System (ADS)
Yao, Ruo-Xia; Wang, Wei; Chen, Ting-Hua
2014-11-01
Motivated by the widely used ansätz method and starting from the modified Riemann—Liouville derivative together with a fractional complex transformation that can be utilized to transform nonlinear fractional partial differential equations to nonlinear ordinary differential equations, new types of exact traveling wave solutions to three important nonlinear space- and time-fractional partial differential equations are obtained simultaneously in terms of solutions of a Riccati equation. The results are new and first reported in this paper.
Belmonte-Beitia, J.; Cuevas, J.
2011-03-15
In this paper, we give a proof of the existence of stationary dark soliton solutions or heteroclinic orbits of nonlinear equations of Schroedinger type with periodic inhomogeneous nonlinearity. The result is illustrated with examples of dark solitons for cubic and photorefractive nonlinearities.
NASA Astrophysics Data System (ADS)
Zecca, Antonio
2016-02-01
The Dirac equation with nonlinear terms induced by torsion is studied in Robertson-Walker (RW) space-time. An extension of a separation method of the equation, based on the Newman-Penrose formalism and previously applied to the nonlinear case, is considered. Accordingly the angular dependence of the Dirac spinor solution is separated, under a special assumption, in the general time-dependent RW metric. In the case of static RW metric the time dependence of the Dirac spinor factors out and one is left with a pair of two coupled nonlinear radial equations. The radial equations are disentangled by a suitable substitution of the spinor solution. The problem amounts then to the solution of a single second-order highly nonlinear differential equation. Some elementary considerations are done on the asymptotic behavior of the solution of the equation.
Numerical solution of the nonlinear Helmholtz equation using nonorthogonal expansions
Fibich, G. . E-mail: fibich@math.tau.ac.il; Tsynkov, S. . E-mail: tsynkov@math.ncsu.edu
2005-11-20
In [J. Comput. Phys. 171 (2001) 632-677] we developed a fourth-order numerical method for solving the nonlinear Helmholtz equation which governs the propagation of time-harmonic laser beams in media with a Kerr-type nonlinearity. A key element of the algorithm was a new nonlocal two-way artificial boundary condition (ABC), set in the direction of beam propagation. This two-way ABC provided for reflectionless propagation of the outgoing waves while also fully transmitting the given incoming beam at the boundaries of the computational domain. Altogether, the algorithm of [J. Comput. Phys. 171 (2001) 632-677] has allowed for a direct simulation of nonlinear self-focusing without neglecting nonparaxial effects and backscattering. To the best of our knowledge, this capacity has never been achieved previously in nonlinear optics. In the current paper, we propose an improved version of the algorithm. The principal innovation is that instead of using the Dirichlet boundary conditions in the direction orthogonal to that of the laser beam propagation, we now introduce Sommerfeld-type local radiation boundary conditions, which are constructed directly in the discrete framework. Numerically, implementation of the Sommerfeld conditions requires evaluation of eigenvalues and eigenvectors for a non-Hermitian matrix. Subsequently, the separation of variables, which is a key building block of the aforementioned nonlocal ABC, is implemented through an expansion with respect to the nonorthogonal basis of the eigenvectors. Numerical simulations show that the new algorithm offers a considerable improvement in its numerical performance, as well as in the range of physical phenomena that it is capable of simulating.
NASA Astrophysics Data System (ADS)
Hopkins, James; Gaudette, Jamie; Mehta, Priyanth
2013-10-01
With the advent of digital signal processing (DSP) in optical transmitters and receivers, the ability to finely tune the ratio of pre and post dispersion compensation can be exploited to best mitigate the nonlinear penalties caused by the Kerr effect. A portion of the nonlinear penalty in optical communication channels has been explained by an increase in peak to average power ratio (PAPR) inherent in highly dispersed signals. The standard approach for minimizing these impairments applies 50% pre dispersion compensation and 50% post dispersion compensation, thereby decreasing average PAPR along the length of the cable, as compared with either 100% pre or post dispersion compensation. In this paper we demonstrate that simply considering the net accumulated dispersion, and applying 50/50 pre/post dispersion is not necessarily the best way to minimize PAPR and subsequent Kerr nonlinearities. Instead, we consider the cumulative dispersion along the entire length of the cable, and, taking into account this additional information, derive an analytic formula for the minimization of PAPR. Alignment with simulation and experimental measurements is presented using a commercially available 100Gb/s dual-polarization binary phase-shift-keying (DP-BPSK) coherent modem, with transmitter and receiver DSP. Measurements are provided from two different 5000km dispersion managed Submarine test-beds, as well as a 3800km terrestrial test-bed with a mixture of SMF-28 and TWRS optical fiber. This method is shown to deviate significantly from the conventional 50/50 method described above, in dispersion managed communications systems, and more closely aligns with results obtained from simulation and data collected from laboratory test-beds.