General method of solving the Schroedinger equation of atoms and molecules

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

Nakatsuji, Hiroshi

2005-12-15

We propose a general method of solving the Schroedinger equation of atoms and molecules. We first construct the wave function having the exact structure, using the ICI (iterative configuration or complement interaction) method and then optimize the variables involved by the variational principle. Based on the scaled Schroedinger equation and related principles, we can avoid the singularity problem of atoms and molecules and formulate a general method of calculating the exact wave functions in an analytical expansion form. We choose initial function {psi}{sub 0} and scaling g function, and then the ICI method automatically generates the wave function that hasmore » the exact structure by using the Hamiltonian of the system. The Hamiltonian contains all the information of the system. The free ICI method provides a flexible and variationally favorable procedure of constructing the exact wave function. We explain the computational procedure of the analytical ICI method routinely performed in our laboratory. Simple examples are given using hydrogen atom for the nuclear singularity case, the Hooke's atom for the electron singularity case, and the helium atom for both cases.« less

Scattering transform for nonstationary Schroedinger equation with bidimensionally perturbed N-soliton potential

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

Boiti, M.; Pempinelli, F.; Pogrebkov, A. K.

2006-12-15

In the framework of the extended resolvent approach the direct and inverse scattering problems for the nonstationary Schroedinger equation with a potential being a perturbation of the N-soliton potential by means of a generic bidimensional smooth function decaying at large spaces are introduced and investigated. The initial value problem of the Kadomtsev-Petviashvili I equation for a solution describing N wave solitons on a generic smooth decaying background is then linearized, giving the time evolution of the spectral data.

Dark soliton solution of Sasa-Satsuma equation

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

Ohta, Y.

2010-03-08

The Sasa-Satsuma equation is a higher order nonlinear Schroedinger type equation which admits bright soliton solutions with internal freedom. We present the dark soliton solutions for the equation by using Gram type determinant. The dark solitons have no internal freedom and exist for both defocusing and focusing equations.

Some Exact Results for the Schroedinger Wave Equation with a Time Dependent Potential

NASA Technical Reports Server (NTRS)

Campbell, Joel

2009-01-01

The time dependent Schroedinger equation with a time dependent delta function potential is solved exactly for many special cases. In all other cases the problem can be reduced to an integral equation of the Volterra type. It is shown that by knowing the wave function at the origin, one may derive the wave function everywhere. Thus, the problem is reduced from a PDE in two variables to an integral equation in one. These results are used to compare adiabatic versus sudden changes in the potential. It is shown that adiabatic changes in the p otential lead to conservation of the normalization of the probability density.

Schroedinger's Wave Structure of Matter (WSM)

NASA Astrophysics Data System (ADS)

Wolff, Milo; Haselhurst, Geoff

2009-10-01

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

Blow-up profile to the solutions of two-coupled Schroedinger equations

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

Chen Jianqing; Guo Boling

2009-02-15

The model of the following two-coupled Schroedinger equations, i{sub t}+(1/2){delta}u=(g{sub 11}|u|{sup 2p}+g|u|{sup p-1}|v|{sup p+1})uu, (t,x)(set-membership sign)R{sub +}xR{sup N}, and iv{sub t}+(1/2){delta}v=(g|u|{sup p+1}|v|{sup p-1}+g{sub 22}|v|{sup 2p})v, (t,x)(set-membership sign)R{sub +}xR{sup N}, is proposed in the study of the Bose-Einstein condensates [Mitchell, et al., ''Self-traping of partially spatially incoherent light,'' Phys. Rev. Lett. 77, 490 (1996)]. We prove that for suitable initial data and p the solution blows up exactly like {delta} function. As a by-product, we prove that similar phenomenon occurs for the critical two-coupled Schroedinger equations with harmonic potential [Perez-Garcia, V. M. and Beitia, T. B., ''Sybiotic solitons in heteronuclear multicomponentmore » Bose-Einstein condensates,'' Phys. Rev. A 72, 033620 (2005)], iu{sub t}+(1/2){delta}u=({omega}/2)|x|{sup 2}u+(g{sub 11}|u|{sup 2p}+g|u|{sup p-1}|v|{sup p+1})u, x(set-membership sign)R{sup N}, and iv{sub t}+(1/2){delta}v=({omega}/2)|x|{sup 2}v+(g|u|{sup p+1}|v|{sup p-1}+g{sub 22}|v|{sup 2p})v, x(set-membership sign)R{sup N}.« less

Linear and nonlinear propagation of water wave groups

NASA Technical Reports Server (NTRS)

Pierson, W. J., Jr.; Donelan, M. A.; Hui, W. H.

1992-01-01

Results are presented from a study of the evolution of waveforms with known analytical group shapes, in the form of both transient wave groups and the cloidal (cn) and dnoidal (dn) wave trains as derived from the nonlinear Schroedinger equation. The waveforms were generated in a long wind-wave tank of the Canada Centre for Inland Waters. It was found that the low-amplitude transients behaved as predicted by the linear theory and that the cn and dn wave trains of moderate steepness behaved almost as predicted by the nonlinear Schroedinger equation. Some of the results did not fit into any of the available theories for waves on water, but they provide important insight on how actual groups of waves propagate and on higher-order effects for a transient waveform.

The Universe according to Schroedinger and Milo

NASA Astrophysics Data System (ADS)

Wolff, Milo

2009-10-01

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

The thermal-wave model: A Schroedinger-like equation for charged particle beam dynamics

NASA Technical Reports Server (NTRS)

Fedele, Renato; Miele, G.

1994-01-01

We review some results on longitudinal beam dynamics obtained in the framework of the Thermal Wave Model (TWM). In this model, which has recently shown the capability to describe both longitudinal and transverse dynamics of charged particle beams, the beam dynamics is ruled by Schroedinger-like equations for the beam wave functions, whose squared modulus is proportional to the beam density profile. Remarkably, the role of the Planck constant is played by a diffractive constant epsilon, the emittance, which has a thermal nature.

Schroedinger's Wave Structure of Matter (WSM)

NASA Astrophysics Data System (ADS)

Wolff, Milo

2009-05-01

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

Nonlinear Gyro-Landau-Fluid Equations

NASA Astrophysics Data System (ADS)

Raskolnikov, I.; Mattor, Nathan; Parker, Scott E.

1996-11-01

We present fluid equations which describe the effects of both linear and nonlinear Landau damping (wave-particle-wave effects). These are derived using a recently developed analytical method similar to renormalization group theory. (Scott E. Parker and Daniele Carati, Phys. Rev. Lett. 75), 441 (1995). In this technique, the phase space structure inherent in Landau damping is treated analytically by building a ``renormalized collisionality'' onto a bare collisionality (which may be taken as vanishingly small). Here we apply this technique to the nonlinear ion gyrokinetic equation in slab geometry, obtaining nonlinear fluid equations for density, parallel momentum and heat. Wave-particle resonances are described by two functions appearing in the heat equation: a renormalized ``collisionality'' and a renormalized nonlinear coupling coeffient. It will be shown that these new equations may correct a deficiency in existing gyrofluid equations, (G. W. Hammett and F. W. Perkins, Phys. Rev. Lett. 64,) 3019 (1990). which can severely underestimate the strength of nonlinear interaction in regimes where linear resonance is strong. (N. Mattor, Phys. Fluids B 4,) 3952 (1992).

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.

Nonlinear differential equations

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

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 ismore » 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.« less

Position dependent mass Schroedinger equation and isospectral potentials: Intertwining operator approach

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

Midya, Bikashkali; Roy, B.; Roychoudhury, R.

2010-02-15

Here, we have studied first- and second-order intertwining approaches to generate isospectral partner potentials of position dependent (effective) mass Schroedinger equation. The second-order intertwiner is constructed directly by taking it as second-order linear differential operator with position dependent coefficients, and the system of equations arising from the intertwining relationship is solved for the coefficients by taking an ansatz. A complete scheme for obtaining general solution is obtained, which is valid for any arbitrary potential and mass function. The proposed technique allows us to generate isospectral potentials with the following spectral modifications: (i) to add new bound state(s), (ii) to removemore » bound state(s), and (iii) to leave the spectrum unaffected. To explain our findings with the help of an illustration, we have used point canonical transformation to obtain the general solution of the position dependent mass Schrodinger equation corresponding to a potential and mass function. It is shown that our results are consistent with the formulation of type A N-fold supersymmetry [T. Tanaka, J. Phys. A 39, 219 (2006); A. Gonzalez-Lopez and T. Tanaka, J. Phys. A 39, 3715 (2006)] for the particular cases N=1 and N=2, respectively.« less

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.

Solving the Schroedinger Equation of Atoms and Molecules without Analytical Integration Based on the Free Iterative-Complement-Interaction Wave Function

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

Nakatsuji, H.; Nakashima, H.; Department of Synthetic Chemistry and Biological Chemistry, Graduate School of Engineering, Kyoto University, Nishikyo-ku, Kyoto 615-8510

2007-12-14

A local Schroedinger equation (LSE) method is proposed for solving the Schroedinger equation (SE) of general atoms and molecules without doing analytic integrations over the complement functions of the free ICI (iterative-complement-interaction) wave functions. Since the free ICI wave function is potentially exact, we can assume a flatness of its local energy. The variational principle is not applicable because the analytic integrations over the free ICI complement functions are very difficult for general atoms and molecules. The LSE method is applied to several 2 to 5 electron atoms and molecules, giving an accuracy of 10{sup -5} Hartree in total energy.more » The potential energy curves of H{sub 2} and LiH molecules are calculated precisely with the free ICI LSE method. The results show the high potentiality of the free ICI LSE method for developing accurate predictive quantum chemistry with the solutions of the SE.« less

Nonlinear Poisson Equation for Heterogeneous Media

PubMed Central

Hu, Langhua; Wei, Guo-Wei

2012-01-01

The Poisson equation is a widely accepted model for electrostatic analysis. However, the Poisson equation is derived based on electric polarizations in a linear, isotropic, and homogeneous dielectric medium. This article introduces a nonlinear Poisson equation to take into consideration of hyperpolarization effects due to intensive charges and possible nonlinear, anisotropic, and heterogeneous media. Variational principle is utilized to derive the nonlinear Poisson model from an electrostatic energy functional. To apply the proposed nonlinear Poisson equation for the solvation analysis, we also construct a nonpolar solvation energy functional based on the nonlinear Poisson equation by using the geometric measure theory. At a fixed temperature, the proposed nonlinear Poisson theory is extensively validated by the electrostatic analysis of the Kirkwood model and a set of 20 proteins, and the solvation analysis of a set of 17 small molecules whose experimental measurements are also available for a comparison. Moreover, the nonlinear Poisson equation is further applied to the solvation analysis of 21 compounds at different temperatures. Numerical results are compared to theoretical prediction, experimental measurements, and those obtained from other theoretical methods in the literature. A good agreement between our results and experimental data as well as theoretical results suggests that the proposed nonlinear Poisson model is a potentially useful model for electrostatic analysis involving hyperpolarization effects. PMID:22947937

Nonlinear Poisson equation for heterogeneous media.

PubMed

Hu, Langhua; Wei, Guo-Wei

2012-08-22

The Poisson equation is a widely accepted model for electrostatic analysis. However, the Poisson equation is derived based on electric polarizations in a linear, isotropic, and homogeneous dielectric medium. This article introduces a nonlinear Poisson equation to take into consideration of hyperpolarization effects due to intensive charges and possible nonlinear, anisotropic, and heterogeneous media. Variational principle is utilized to derive the nonlinear Poisson model from an electrostatic energy functional. To apply the proposed nonlinear Poisson equation for the solvation analysis, we also construct a nonpolar solvation energy functional based on the nonlinear Poisson equation by using the geometric measure theory. At a fixed temperature, the proposed nonlinear Poisson theory is extensively validated by the electrostatic analysis of the Kirkwood model and a set of 20 proteins, and the solvation analysis of a set of 17 small molecules whose experimental measurements are also available for a comparison. Moreover, the nonlinear Poisson equation is further applied to the solvation analysis of 21 compounds at different temperatures. Numerical results are compared to theoretical prediction, experimental measurements, and those obtained from other theoretical methods in the literature. A good agreement between our results and experimental data as well as theoretical results suggests that the proposed nonlinear Poisson model is a potentially useful model for electrostatic analysis involving hyperpolarization effects. Copyright © 2012 Biophysical Society. Published by Elsevier Inc. All rights reserved.

Explicit blow-up solutions to the Schroedinger maps from R{sup 2} to the hyperbolic 2-space H{sup 2}

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

Ding Qing

2009-10-15

In this article, we prove that the equation of the Schroedinger maps from R{sup 2} to the hyperbolic 2-space H{sup 2} is SU(1,1)-gauge equivalent to the following 1+2 dimensional nonlinear Schroedinger-type system of three unknown complex functions p, q, r, and a real function u: iq{sub t}+q{sub zz}-2uq+2(pq){sub z}-2pq{sub z}-4|p|{sup 2}q=0, ir{sub t}-r{sub zz}+2ur+2(pr){sub z}-2pr{sub z}+4|p|{sup 2}r=0, ip{sub t}+(qr){sub z}-u{sub z}=0, p{sub z}+p{sub z}=-|q|{sup 2}+|r|{sup 2}, -r{sub z}+q{sub z}=-2(pr+pq), where z is a complex coordinate of the plane R{sup 2} and z is the complex conjugate of z. Although this nonlinear Schroedinger-type system looks complicated, it admits a class ofmore » explicit blow-up smooth solutions: p=0, q=(e{sup i(bzz/2(a+bt))}/a+bt){alpha}z, r=e{sup -i(bzz/2(a+bt))}/(a+bt){alpha}z, u=2{alpha}{sup 2}zz/(a+bt){sup 2}, where a and b are real numbers with ab<0 and {alpha} satisfies {alpha}{sup 2}=b{sup 2}/16. From these facts, we explicitly construct smooth solutions to the Schroedinger maps from R{sup 2} to the hyperbolic 2-space H{sup 2} by using the gauge transformations such that the absolute values of their gradients blow up in finite time. This reveals some blow-up phenomenon of Schroedinger maps.« less

Amplification of nonlinear surface waves by wind

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

Leblanc, Stephane

2007-10-15

A weakly nonlinear analysis is conducted to study the evolution of slowly varying wavepackets with small but finite amplitudes, that evolve at the interface between air and water under the effect of wind. In the inviscid assumption, wave envelopes are governed by cubic nonlinear Schroedinger or Davey-Stewartson equations forced by a linear term corresponding to Miles' mechanism of wave generation. Under fair wind, it is shown that Stokes waves grow exponentially and that Benjamin-Feir instability becomes explosive.

A numerical scheme for nonlinear Helmholtz equations with strong nonlinear optical effects.

PubMed

Xu, Zhengfu; Bao, Gang

2010-11-01

A numerical scheme is presented to solve the nonlinear Helmholtz (NLH) equation modeling second-harmonic generation (SHG) in photonic bandgap material doped with a nonlinear χ((2)) effect and the NLH equation modeling wave propagation in Kerr type gratings with a nonlinear χ((3)) effect in the one-dimensional case. Both of these nonlinear phenomena arise as a result of the combination of high electromagnetic mode density and nonlinear reaction from the medium. When the mode intensity of the incident wave is significantly strong, which makes the nonlinear effect non-negligible, numerical methods based on the linearization of the essentially nonlinear problem will become inadequate. In this work, a robust, stable numerical scheme is designed to simulate the NLH equations with strong nonlinearity.

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.

Modified Method of Simplest Equation Applied to the Nonlinear Schrödinger Equation

NASA Astrophysics Data System (ADS)

Vitanov, Nikolay K.; Dimitrova, Zlatinka I.

2018-03-01

We consider an extension of the methodology of the modified method of simplest equation to the case of use of two simplest equations. The extended methodology is applied for obtaining exact solutions of model nonlinear partial differential equations for deep water waves: the nonlinear Schrödinger equation. It is shown that the methodology works also for other equations of the nonlinear Schrödinger kind.

Solving the electron and electron-nuclear Schroedinger equations for the excited states of helium atom with the free iterative-complement-interaction method

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

Nakashima, Hiroyuki; Hijikata, Yuh; Nakatsuji, Hiroshi

2008-04-21

Very accurate variational calculations with the free iterative-complement-interaction (ICI) method for solving the Schroedinger equation were performed for the 1sNs singlet and triplet excited states of helium atom up to N=24. This is the first extensive applications of the free ICI method to the calculations of excited states to very high levels. We performed the calculations with the fixed-nucleus Hamiltonian and moving-nucleus Hamiltonian. The latter case is the Schroedinger equation for the electron-nuclear Hamiltonian and includes the quantum effect of nuclear motion. This solution corresponds to the nonrelativistic limit and reproduced the experimental values up to five decimal figures. Themore » small differences from the experimental values are not at all the theoretical errors but represent the physical effects that are not included in the present calculations, such as relativistic effect, quantum electrodynamic effect, and even the experimental errors. The present calculations constitute a small step toward the accurately predictive quantum chemistry.« less

Oscillating solutions for nonlinear Helmholtz equations

NASA Astrophysics Data System (ADS)

Mandel, Rainer; Montefusco, Eugenio; Pellacci, Benedetta

2017-12-01

Existence results for radially symmetric oscillating solutions for a class of nonlinear autonomous Helmholtz equations are given and their exact asymptotic behaviour at infinity is established. Some generalizations to nonautonomous radial equations as well as existence results for nonradial solutions are found. Our theorems prove the existence of standing waves solutions of nonlinear Klein-Gordon or Schrödinger equations with large frequencies.

Fluid equations with nonlinear wave-particle resonances^

NASA Astrophysics Data System (ADS)

Mattor, Nathan

1997-11-01

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

Distribution-valued initial data for the complex Ginzburg-Landau equation

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

Levermore, C.D.; Oliver, M.

1997-11-01

The generalized complex Ginzburg-Landau (CGL) equation with a nonlinearity of order 2{sigma} + 1 in d spatial dimensions has a unique local classical solution for distributional initial data in the Sobolev space H{sup q} provided that q > d/2 - 1/{sigma}. This result directly corresponds to a theorem for the nonlinear Schroedinger (NLS) equation which has been proved by Cazenave and Weissler in 1990. While the proof in the NLS case relies on Besov space techniques, it is shown here that for the CGL equation, the smoothing properties of the linear semigroup can be eased to obtain an almost optimalmore » result by elementary means. 1 fig.« less

Solving the Schroedinger equation for helium atom and its isoelectronic ions with the free iterative complement interaction (ICI) method

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

Nakashima, Hiroyuki; Nakatsuji, Hiroshi

2007-12-14

The Schroedinger equation was solved very accurately for helium atom and its isoelectronic ions (Z=1-10) with the free iterative complement interaction (ICI) method followed by the variational principle. We obtained highly accurate wave functions and energies of helium atom and its isoelectronic ions. For helium, the calculated energy was -2.903 724 377 034 119 598 311 159 245 194 404 446 696 905 37 a.u., correct over 40 digit accuracy, and for H{sup -}, it was -0.527 751 016 544 377 196 590 814 566 747 511 383 045 02 a.u. These results prove numerically that with the free ICImore » method, we can calculate the solutions of the Schroedinger equation as accurately as one desires. We examined several types of scaling function g and initial function {psi}{sub 0} of the free ICI method. The performance was good when logarithm functions were used in the initial function because the logarithm function is physically essential for three-particle collision area. The best performance was obtained when we introduce a new logarithm function containing not only r{sub 1} and r{sub 2} but also r{sub 12} in the same logarithm function.« less

Solutions of the cylindrical nonlinear Maxwell equations.

PubMed

Xiong, Hao; Si, Liu-Gang; Ding, Chunling; Lü, Xin-You; Yang, Xiaoxue; Wu, Ying

2012-01-01

Cylindrical nonlinear optics is a burgeoning research area which describes cylindrical electromagnetic wave propagation in nonlinear media. Finding new exact solutions for different types of nonlinearity and inhomogeneity to describe cylindrical electromagnetic wave propagation is of great interest and meaningful for theory and application. This paper gives exact solutions for the cylindrical nonlinear Maxwell equations and presents an interesting connection between the exact solutions for different cylindrical nonlinear Maxwell equations. We also provide some examples and discussion to show the application of the results we obtained. Our results provide the basis for solving complex systems of nonlinearity and inhomogeneity with simple systems.

Robust iterative method for nonlinear Helmholtz equation

NASA Astrophysics Data System (ADS)

Yuan, Lijun; Lu, Ya Yan

2017-08-01

A new iterative method is developed for solving the two-dimensional nonlinear Helmholtz equation which governs polarized light in media with the optical Kerr nonlinearity. In the strongly nonlinear regime, the nonlinear Helmholtz equation could have multiple solutions related to phenomena such as optical bistability and symmetry breaking. The new method exhibits a much more robust convergence behavior than existing iterative methods, such as frozen-nonlinearity iteration, Newton's method and damped Newton's method, and it can be used to find solutions when good initial guesses are unavailable. Numerical results are presented for the scattering of light by a nonlinear circular cylinder based on the exact nonlocal boundary condition and a pseudospectral method in the polar coordinate system.

The soliton transform and a possible application to nonlinear Alfven waves in space

NASA Technical Reports Server (NTRS)

Hada, T.; Hamilton, R. L.; Kennel, C. F.

1993-01-01

The inverse scattering transform (IST) based on the derivative nonlinear Schroedinger (DNLS) equation is applied to a complex time series of nonlinear Alfven wave data generated by numerical simulation. The IST describes the long-time evolution of quasi-parallel Alfven waves more efficiently than the Fourier transform, which is adapted to linear rather than nonlinear problems. When dissipation is added, so the conditions for the validity of the DNLS are not strictly satisfied, the IST continues to provide a compact description of the wavefield in terms of a small number of decaying envelope solitons.

Convergence of Galerkin approximations for operator Riccati equations: A nonlinear evolution equation approach

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.

Integrability and structural stability of solutions to the Ginzburg-Landau equation

NASA Technical Reports Server (NTRS)

Keefe, Laurence R.

1986-01-01

The integrability of the Ginzburg-Landau equation is studied to investigate if the existence of chaotic solutions found numerically could have been predicted a priori. The equation is shown not to possess the Painleveproperty, except for a special case of the coefficients that corresponds to the integrable, nonlinear Schroedinger (NLS) equation. Regarding the Ginzburg-Landau equation as a dissipative perturbation of the NLS, numerical experiments show all but one of a family of two-tori solutions, possessed by the NLS under particular conditions, to disappear under real perturbations to the NLS coefficients of O(10 to the -6th).

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

PubMed

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

2013-07-01

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

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.

Entropy production and nonlinear Fokker-Planck equations.

PubMed

Casas, G A; Nobre, F D; Curado, E M F

2012-12-01

The entropy time rate of systems described by nonlinear Fokker-Planck equations--which are directly related to generalized entropic forms--is analyzed. Both entropy production, associated with irreversible processes, and entropy flux from the system to its surroundings are studied. Some examples of known generalized entropic forms are considered, and particularly, the flux and production of the Boltzmann-Gibbs entropy, obtained from the linear Fokker-Planck equation, are recovered as particular cases. Since nonlinear Fokker-Planck equations are appropriate for the dynamical behavior of several physical phenomena in nature, like many within the realm of complex systems, the present analysis should be applicable to irreversible processes in a large class of nonlinear systems, such as those described by Tsallis and Kaniadakis entropies.

Exact traveling wave solutions for system of nonlinear evolution equations.

PubMed

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

2016-01-01

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

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.

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.

Green functions and Langevin equations for nonlinear diffusion equations: A comment on ‘Markov processes, Hurst exponents, and nonlinear diffusion equations’ by Bassler et al.

NASA Astrophysics Data System (ADS)

Frank, T. D.

2008-02-01

We discuss two central claims made in the study by Bassler et al. [K.E. Bassler, G.H. Gunaratne, J.L. McCauley, Physica A 369 (2006) 343]. Bassler et al. claimed that Green functions and Langevin equations cannot be defined for nonlinear diffusion equations. In addition, they claimed that nonlinear diffusion equations are linear partial differential equations disguised as nonlinear ones. We review bottom-up and top-down approaches that have been used in the literature to derive Green functions for nonlinear diffusion equations and, in doing so, show that the first claim needs to be revised. We show that the second claim as well needs to be revised. To this end, we point out similarities and differences between non-autonomous linear Fokker-Planck equations and autonomous nonlinear Fokker-Planck equations. In this context, we raise the question whether Bassler et al.’s approach to financial markets is physically plausible because it necessitates the introduction of external traders and causes. Such external entities can easily be eliminated when taking self-organization principles and concepts of nonextensive thermostatistics into account and modeling financial processes by means of nonlinear Fokker-Planck equations.

A parallel algorithm for nonlinear convection-diffusion equations

NASA Technical Reports Server (NTRS)

Scroggs, Jeffrey S.

1990-01-01

A parallel algorithm for the efficient solution of nonlinear time-dependent convection-diffusion equations with small parameter on the diffusion term is presented. The method is based on a physically motivated domain decomposition that is dictated by singular perturbation analysis. The analysis is used to determine regions where certain reduced equations may be solved in place of the full equation. The method is suitable for the solution of problems arising in the simulation of fluid dynamics. Experimental results for a nonlinear equation in two-dimensions are presented.

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.

Nonlinear equations of dynamics for spinning paraboloidal antennas

NASA Technical Reports Server (NTRS)

Utku, S.; Shoemaker, W. L.; Salama, M.

1983-01-01

The nonlinear strain-displacement and velocity-displacement relations of spinning imperfect rotational paraboloidal thin shell antennas are derived for nonaxisymmetrical deformations. Using these relations with the admissible trial functions in the principle functional of dynamics, the nonlinear equations of stress inducing motion are expressed in the form of a set of quasi-linear ordinary differential equations of the undetermined functions by means of the Rayleigh-Ritz procedure. These equations include all nonlinear terms up to and including the third degree. Explicit expressions are given for the coefficient matrices appearing in these equations. Both translational and rotational off-sets of the axis of revolution (and also the apex point of the paraboloid) with respect to the spin axis are considered. Although the material of the antenna is assumed linearly elastic, it can be anisotropic.

Nonlinear and dissipative constitutive equations for coupled first-order acoustic field equations that are consistent with the generalized Westervelt equation

NASA Astrophysics Data System (ADS)

Verweij, Martin D.; Huijssen, Jacob

2006-05-01

In diagnostic medical ultrasound, it has become increasingly important to evaluate the nonlinear field of an acoustic beam that propagates in a weakly nonlinear, dissipative medium and that is steered off-axis up to very wide angles. In this case, computations cannot be based on the widely used KZK equation since it applies only to small angles. To benefit from successful computational schemes from elastodynamics and electromagnetics, we propose to use two first-order acoustic field equations, accompanied by two constitutive equations, as an alternative basis. This formulation quite naturally results in the contrast source formalism, makes a clear distinction between fundamental conservation laws and medium behavior, and allows for a straightforward inclusion of any medium inhomogenities. This paper is concerned with the derivation of relevant constitutive equations. We take a pragmatic approach and aim to find those constitutive equations that represent the same medium as implicitly described by the recognized, full wave, nonlinear equations such as the generalized Westervelt equation. We will show how this is achieved by considering the nonlinear case without attenuation, the linear case with attenuation, and the nonlinear case with attenuation. As a result we will obtain surprisingly simple constitutive equations for the full wave case.

Modulational-instability-induced supercontinuum generation with saturable nonlinear response

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

Raja, R. Vasantha Jayakantha; Porsezian, K.; Nithyanandan, K.

2010-07-15

We theoretically investigate the supercontinuum generation (SCG) on the basis of modulational instability (MI) in liquid-core photonic crystal fibers (LCPCF) with CS{sub 2}-filled central core. The effect of saturable nonlinearity of LCPCF on SCG in the femtosecond regime is studied using an appropriately modified nonlinear Schroedinger equation. We also compare the MI induced spectral broadening with SCG obtained by soliton fission. To analyze the quality of the pulse broadening, we study the coherence of the SC pulse numerically. It is evident from the numerical simulation that the response of the saturable nonlinearity suppresses the broadening of the pulse. We alsomore » observe that the MI induced SCG in the presence of saturable nonlinearity degrades the coherence of the SCG pulse when compared to unsaturated medium.« less

Proposed solution methodology for the dynamically coupled nonlinear geared rotor mechanics equations

NASA Technical Reports Server (NTRS)

Mitchell, L. D.; David, J. W.

1983-01-01

The equations which describe the three-dimensional motion of an unbalanced rigid disk in a shaft system are nonlinear and contain dynamic-coupling terms. Traditionally, investigators have used an order analysis to justify ignoring the nonlinear terms in the equations of motion, producing a set of linear equations. This paper will show that, when gears are included in such a rotor system, the nonlinear dynamic-coupling terms are potentially as large as the linear terms. Because of this, one must attempt to solve the nonlinear rotor mechanics equations. A solution methodology is investigated to obtain approximate steady-state solutions to these equations. As an example of the use of the technique, a simpler set of equations is solved and the results compared to numerical simulations. These equations represent the forced, steady-state response of a spring-supported pendulum. These equations were chosen because they contain the type of nonlinear terms found in the dynamically-coupled nonlinear rotor equations. The numerical simulations indicate this method is reasonably accurate even when the nonlinearities are large.

Superposition of elliptic functions as solutions for a large number of nonlinear equations

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

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 asmore » 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.« less

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

Analytical solutions of the Schroedinger equation for a two-dimensional exciton in magnetic field of arbitrary strength

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

Hoang-Do, Ngoc-Tram; Hoang, Van-Hung; Le, Van-Hoang

2013-05-15

The Feranchuk-Komarov operator method is developed by combining with the Levi-Civita transformation in order to construct analytical solutions of the Schroedinger equation for a two-dimensional exciton in a uniform magnetic field of arbitrary strength. As a result, analytical expressions for the energy of the ground and excited states are obtained with a very high precision of up to four decimal places. Especially, the precision is uniformly stable for the whole range of the magnetic field. This advantage appears due to the consideration of the asymptotic behaviour of the wave-functions in strong magnetic field. The results could be used for variousmore » physical analyses and the method used here could also be applied to other atomic systems.« less

Entropy and convexity for nonlinear partial differential equations

PubMed Central

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

Entropy and convexity for nonlinear partial differential equations.

PubMed

Ball, John M; Chen, Gui-Qiang G

2013-12-28

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.

Lipschitz regularity results for nonlinear strictly elliptic equations and applications

NASA Astrophysics Data System (ADS)

Ley, Olivier; Nguyen, Vinh Duc

2017-10-01

Most of Lipschitz regularity results for nonlinear strictly elliptic equations are obtained for a suitable growth power of the nonlinearity with respect to the gradient variable (subquadratic for instance). For equations with superquadratic growth power in gradient, one usually uses weak Bernstein-type arguments which require regularity and/or convex-type assumptions on the gradient nonlinearity. In this article, we obtain new Lipschitz regularity results for a large class of nonlinear strictly elliptic equations with possibly arbitrary growth power of the Hamiltonian with respect to the gradient variable using some ideas coming from Ishii-Lions' method. We use these bounds to solve an ergodic problem and to study the regularity and the large time behavior of the solution of the evolution equation.

Solitary wave for a nonintegrable discrete nonlinear Schrödinger equation in nonlinear optical waveguide arrays

NASA Astrophysics Data System (ADS)

Ma, Li-Yuan; Ji, Jia-Liang; Xu, Zong-Wei; Zhu, Zuo-Nong

2018-03-01

We study a nonintegrable discrete nonlinear Schrödinger (dNLS) equation with the term of nonlinear nearest-neighbor interaction occurred in nonlinear optical waveguide arrays. By using discrete Fourier transformation, we obtain numerical approximations of stationary and travelling solitary wave solutions of the nonintegrable dNLS equation. The analysis of stability of stationary solitary waves is performed. It is shown that the nonlinear nearest-neighbor interaction term has great influence on the form of solitary wave. The shape of solitary wave is important in the electric field propagating. If we neglect the nonlinear nearest-neighbor interaction term, much important information in the electric field propagating may be missed. Our numerical simulation also demonstrates the difference of chaos phenomenon between the nonintegrable dNLS equation with nonlinear nearest-neighbor interaction and another nonintegrable dNLS equation without the term. Project supported by the National Natural Science Foundation of China (Grant Nos. 11671255 and 11701510), the Ministry of Economy and Competitiveness of Spain (Grant No. MTM2016-80276-P (AEI/FEDER, EU)), and the China Postdoctoral Science Foundation (Grant No. 2017M621964).

A new solution procedure for a nonlinear infinite beam equation of motion

NASA Astrophysics Data System (ADS)

Jang, T. S.

2016-10-01

Our goal of this paper is of a purely theoretical question, however which would be fundamental in computational partial differential equations: Can a linear solution-structure for the equation of motion for an infinite nonlinear beam be directly manipulated for constructing its nonlinear solution? Here, the equation of motion is modeled as mathematically a fourth-order nonlinear partial differential equation. To answer the question, a pseudo-parameter is firstly introduced to modify the equation of motion. And then, an integral formalism for the modified equation is found here, being taken as a linear solution-structure. It enables us to formulate a nonlinear integral equation of second kind, equivalent to the original equation of motion. The fixed point approach, applied to the integral equation, results in proposing a new iterative solution procedure for constructing the nonlinear solution of the original beam equation of motion, which consists luckily of just the simple regular numerical integration for its iterative process; i.e., it appears to be fairly simple as well as straightforward to apply. A mathematical analysis is carried out on both natures of convergence and uniqueness of the iterative procedure by proving a contractive character of a nonlinear operator. It follows conclusively,therefore, that it would be one of the useful nonlinear strategies for integrating the equation of motion for a nonlinear infinite beam, whereby the preceding question may be answered. In addition, it may be worth noticing that the pseudo-parameter introduced here has double roles; firstly, it connects the original beam equation of motion with the integral equation, second, it is related with the convergence of the iterative method proposed here.

Observation of Quasi-Two-Dimensional Nonlinear Interactions in a Drift-Wave Streamer

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

Yamada, T.; Nagashima, Y.; Itoh, S.-I.

2010-11-26

A streamer, which is a bunching of drift-wave fluctuations, and its mediator, which generates the streamer by coupling with other fluctuations, have been observed in a cylindrical magnetized plasma. Their radial structures were investigated in detail by using the biphase analysis. Their quasi-two-dimensional structures were revealed to be equivalent with a pair of fast and slow modes predicted by a nonlinear Schroedinger equation based on the Hasegawa-Mima model.

Prolongation structures of nonlinear evolution equations

NASA Technical Reports Server (NTRS)

Wahlquist, H. D.; Estabrook, F. B.

1975-01-01

A technique is developed for systematically deriving a 'prolongation structure' - a set of interrelated potentials and pseudopotentials - for nonlinear partial differential equations in two independent variables. When this is applied to the Korteweg-de Vries equation, a new infinite set of conserved quantities is obtained. Known solution techniques are shown to result from the discovery of such a structure: related partial differential equations for the potential functions, linear 'inverse scattering' equations for auxiliary functions, Backlund transformations. Generalizations of these techniques will result from the use of irreducible matrix representations of the prolongation structure.

Kinetic treatment of nonlinear magnetized plasma motions - General geometry and parallel waves

NASA Technical Reports Server (NTRS)

Khabibrakhmanov, I. KH.; Galinskii, V. L.; Verheest, F.

1992-01-01

The expansion of kinetic equations in the limit of a strong magnetic field is presented. This gives a natural description of the motions of magnetized plasmas, which are slow compared to the particle gyroperiods and gyroradii. Although the approach is 3D, this very general result is used only to focus on the parallel propagation of nonlinear Alfven waves. The derivative nonlinear Schroedinger-like equation is obtained. Two new terms occur compared to earlier treatments, a nonlinear term proportional to the heat flux along the magnetic field line and a higher-order dispersive term. It is shown that kinetic description avoids the singularities occurring in magnetohydrodynamic or multifluid approaches, which correspond to the degenerate case of sound speeds equal to the Alfven speed, and that parallel heat fluxes cannot be neglected, not even in the case of low parallel plasma beta. A truly stationary soliton solution is derived.

Self-accelerating self-trapped nonlinear beams of Maxwell's equations.

PubMed

Kaminer, Ido; Nemirovsky, Jonathan; Segev, Mordechai

2012-08-13

We present shape-preserving self-accelerating beams of Maxwell's equations with optical nonlinearities. Such beams are exact solutions to Maxwell's equations with Kerr or saturable nonlinearity. The nonlinearity contributes to self-trapping and causes backscattering. Those effects, together with diffraction effects, work to maintain shape-preserving acceleration of the beam on a circular trajectory. The backscattered beam is found to be a key issue in the dynamics of such highly non-paraxial nonlinear beams. To study that, we develop two new techniques: projection operator separating the forward and backward waves, and reverse simulation. Finally, we discuss the possibility that such beams would reflect themselves through the nonlinear effect, to complete a 'U' shaped trajectory.

Integrable pair-transition-coupled nonlinear Schrödinger equations.

PubMed

Ling, Liming; Zhao, Li-Chen

2015-08-01

We study integrable coupled nonlinear Schrödinger equations with pair particle transition between components. Based on exact solutions of the coupled model with attractive or repulsive interaction, we predict that some new dynamics of nonlinear excitations can exist, such as the striking transition dynamics of breathers, new excitation patterns for rogue waves, topological kink excitations, and other new stable excitation structures. In particular, we find that nonlinear wave solutions of this coupled system can be written as a linear superposition of solutions for the simplest scalar nonlinear Schrödinger equation. Possibilities to observe them are discussed in a cigar-shaped Bose-Einstein condensate with two hyperfine states. The results would enrich our knowledge on nonlinear excitations in many coupled nonlinear systems with transition coupling effects, such as multimode nonlinear fibers, coupled waveguides, and a multicomponent Bose-Einstein condensate system.

The Nonlinear Steepest Descent Method to Long-Time Asymptotics of the Coupled Nonlinear Schrödinger Equation

NASA Astrophysics Data System (ADS)

Geng, Xianguo; Liu, Huan

2018-04-01

The Riemann-Hilbert problem for the coupled nonlinear Schrödinger equation is formulated on the basis of the corresponding 3× 3 matrix spectral problem. Using the nonlinear steepest descent method, we obtain leading-order asymptotics for the Cauchy problem of the coupled nonlinear Schrödinger equation.

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.

The nonlinear modified equation approach to analyzing finite difference schemes

NASA Technical Reports Server (NTRS)

Klopfer, G. H.; Mcrae, D. S.

1981-01-01

The nonlinear modified equation approach is taken in this paper to analyze the generalized Lax-Wendroff explicit scheme approximation to the unsteady one- and two-dimensional equations of gas dynamics. Three important applications of the method are demonstrated. The nonlinear modified equation analysis is used to (1) generate higher order accurate schemes, (2) obtain more accurate estimates of the discretization error for nonlinear systems of partial differential equations, and (3) generate an adaptive mesh procedure for the unsteady gas dynamic equations. Results are obtained for all three areas. For the adaptive mesh procedure, mesh point requirements for equal resolution of discontinuities were reduced by a factor of five for a 1-D shock tube problem solved by the explicit MacCormack scheme.

The Swift-Hohenberg equation with a nonlocal nonlinearity

NASA Astrophysics Data System (ADS)

Morgan, David; Dawes, Jonathan H. P.

2014-03-01

It is well known that aspects of the formation of localised states in a one-dimensional Swift-Hohenberg equation can be described by Ginzburg-Landau-type envelope equations. This paper extends these multiple scales analyses to cases where an additional nonlinear integral term, in the form of a convolution, is present. The presence of a kernel function introduces a new lengthscale into the problem, and this results in additional complexity in both the derivation of envelope equations and in the bifurcation structure. When the kernel is short-range, weakly nonlinear analysis results in envelope equations of standard type but whose coefficients are modified in complicated ways by the nonlinear nonlocal term. Nevertheless, these computations can be formulated quite generally in terms of properties of the Fourier transform of the kernel function. When the lengthscale associated with the kernel is longer, our method leads naturally to the derivation of two different, novel, envelope equations that describe aspects of the dynamics in these new regimes. The first of these contains additional bifurcations, and unexpected loops in the bifurcation diagram. The second of these captures the stretched-out nature of the homoclinic snaking curves that arises due to the nonlocal term.

The backward phase flow and FBI-transform-based Eulerian Gaussian beams for the Schroedinger equation

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

Leung Shingyu, E-mail: masyleung@ust.h; Qian Jianliang, E-mail: qian@math.msu.ed

2010-11-20

We propose the backward phase flow method to implement the Fourier-Bros-Iagolnitzer (FBI)-transform-based Eulerian Gaussian beam method for solving the Schroedinger equation in the semi-classical regime. The idea of Eulerian Gaussian beams has been first proposed in . In this paper we aim at two crucial computational issues of the Eulerian Gaussian beam method: how to carry out long-time beam propagation and how to compute beam ingredients rapidly in phase space. By virtue of the FBI transform, we address the first issue by introducing the reinitialization strategy into the Eulerian Gaussian beam framework. Essentially we reinitialize beam propagation by applying themore » FBI transform to wavefields at intermediate time steps when the beams become too wide. To address the second issue, inspired by the original phase flow method, we propose the backward phase flow method which allows us to compute beam ingredients rapidly. Numerical examples demonstrate the efficiency and accuracy of the proposed algorithms.« less

A new perturbative approach to nonlinear partial differential equations

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

Bender, C.M.; Boettcher, S.; Milton, K.A.

1991-11-01

This paper shows how to solve some nonlinear wave equations as perturbation expansions in powers of a parameter that expresses the degree of nonlinearity. For the case of the Burgers equation {ital u}{sub {ital t}}+{ital uu}{sub {ital x}}={ital u}{sub {ital xx}}, the general nonlinear equation {ital u}{sub {ital t}}+{ital u}{sup {delta}}{ital u}{sub {ital x}}={ital u}{sub {ital xx}} is considered and expanded in powers of {delta}. The coefficients of the {delta} series to sixth order in powers of {delta} is determined and Pade summation is used to evaluate the perturbation series for large values of {delta}. The numerical results are accuratemore » and the method is very general; it applies to other well-studied partial differential equations such as the Korteweg--de Vries equation, {ital u}{sub {ital t}}+{ital uu}{sub {ital x}} ={ital u}{sub {ital xxx}}.« less

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.

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.

Nonlinear grid error effects on numerical solution of partial differential equations

NASA Technical Reports Server (NTRS)

Dey, S. K.

1980-01-01

Finite difference solutions of nonlinear partial differential equations require discretizations and consequently grid errors are generated. These errors strongly affect stability and convergence properties of difference models. Previously such errors were analyzed by linearizing the difference equations for solutions. Properties of mappings of decadence were used to analyze nonlinear instabilities. Such an analysis is directly affected by initial/boundary conditions. An algorithm was developed, applied to nonlinear Burgers equations, and verified computationally. A preliminary test shows that Navier-Stokes equations may be treated similarly.

Hazardous Continuation Backward in Time in Nonlinear Parabolic Equations, and an Experiment in Deblurring Nonlinearly Blurred Imagery

PubMed Central

Carasso, Alfred S

2013-01-01

Identifying sources of ground water pollution, and deblurring nanoscale imagery as well as astronomical galaxy images, are two important applications involving numerical computation of parabolic equations backward in time. Surprisingly, very little is known about backward continuation in nonlinear parabolic equations. In this paper, an iterative procedure originating in spectroscopy in the 1930’s, is adapted into a useful tool for solving a wide class of 2D nonlinear backward parabolic equations. In addition, previously unsuspected difficulties are uncovered that may preclude useful backward continuation in parabolic equations deviating too strongly from the linear, autonomous, self adjoint, canonical model. This paper explores backward continuation in selected 2D nonlinear equations, by creating fictitious blurred images obtained by using several sharp images as initial data in these equations, and capturing the corresponding solutions at some positive time T. Successful backward continuation from t=T to t = 0, would recover the original sharp image. Visual recognition provides meaningful evaluation of the degree of success or failure in the reconstructed solutions. Instructive examples are developed, illustrating the unexpected influence of certain types of nonlinearities. Visually and statistically indistinguishable blurred images are presented, with vastly different deblurring results. These examples indicate that how an image is nonlinearly blurred is critical, in addition to the amount of blur. The equations studied represent nonlinear generalizations of Brownian motion, and the blurred images may be interpreted as visually expressing the results of novel stochastic processes. PMID:26401430

Hazardous Continuation Backward in Time in Nonlinear Parabolic Equations, and an Experiment in Deblurring Nonlinearly Blurred Imagery.

PubMed

Carasso, Alfred S

2013-01-01

Identifying sources of ground water pollution, and deblurring nanoscale imagery as well as astronomical galaxy images, are two important applications involving numerical computation of parabolic equations backward in time. Surprisingly, very little is known about backward continuation in nonlinear parabolic equations. In this paper, an iterative procedure originating in spectroscopy in the 1930's, is adapted into a useful tool for solving a wide class of 2D nonlinear backward parabolic equations. In addition, previously unsuspected difficulties are uncovered that may preclude useful backward continuation in parabolic equations deviating too strongly from the linear, autonomous, self adjoint, canonical model. This paper explores backward continuation in selected 2D nonlinear equations, by creating fictitious blurred images obtained by using several sharp images as initial data in these equations, and capturing the corresponding solutions at some positive time T. Successful backward continuation from t=T to t = 0, would recover the original sharp image. Visual recognition provides meaningful evaluation of the degree of success or failure in the reconstructed solutions. Instructive examples are developed, illustrating the unexpected influence of certain types of nonlinearities. Visually and statistically indistinguishable blurred images are presented, with vastly different deblurring results. These examples indicate that how an image is nonlinearly blurred is critical, in addition to the amount of blur. The equations studied represent nonlinear generalizations of Brownian motion, and the blurred images may be interpreted as visually expressing the results of novel stochastic processes.

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.

GHM method for obtaining rationalsolutions of nonlinear differential equations.

PubMed

Vazquez-Leal, Hector; Sarmiento-Reyes, Arturo

2015-01-01

In this paper, we propose the application of the general homotopy method (GHM) to obtain rational solutions of nonlinear differential equations. It delivers a high precision representation of the nonlinear differential equation using a few linear algebraic terms. In order to assess the benefits of this proposal, three nonlinear problems are solved and compared against other semi-analytic methods or numerical methods. The obtained results show that GHM is a powerful tool, capable to generate highly accurate rational solutions. AMS subject classification 34L30.

Absorbing boundaries in numerical solutions of the time-dependent Schroedinger equation on a grid using exterior complex scaling

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

He, F.; Ruiz, C.; Becker, A.

We study the suppression of reflections in the numerical simulation of the time-dependent Schroedinger equation for strong-field problems on a grid using exterior complex scaling (ECS) as an absorbing boundary condition. It is shown that the ECS method can be applied in both the length and the velocity gauge as long as appropriate approximations are applied in the ECS transformation of the electron-field coupling. It is found that the ECS method improves the suppression of reflection as compared to the conventional masking function technique in typical simulations of atoms exposed to an intense laser pulse. Finally, we demonstrate the advantagemore » of the ECS technique to avoid unphysical artifacts in the evaluation of high harmonic spectra.« less

Analytical solution of the nonlinear diffusion equation

NASA Astrophysics Data System (ADS)

Shanker Dubey, Ravi; Goswami, Pranay

2018-05-01

In the present paper, we derive the solution of the nonlinear fractional partial differential equations using an efficient approach based on the q -homotopy analysis transform method ( q -HATM). The fractional diffusion equations derivatives are considered in Caputo sense. The derived results are graphically demonstrated as well.

Modelling nonlinearity in piezoceramic transducers: From equations to nonlinear equivalent circuits.

PubMed

Parenthoine, D; Tran-Huu-Hue, L-P; Haumesser, L; Vander Meulen, F; Lematre, M; Lethiecq, M

2011-02-01

Quadratic nonlinear equations of a piezoelectric element under the assumptions of 1D vibration and weak nonlinearity are derived by the perturbation theory. It is shown that the nonlinear response can be represented by controlled sources that are added to the classical hexapole used to model piezoelectric ultrasonic transducers. As a consequence, equivalent electrical circuits can be used to predict the nonlinear response of a transducer taking into account the acoustic loads on the rear and front faces. A generalisation of nonlinear equivalent electrical circuits to cases including passive layers and propagation media is then proposed. Experimental results, in terms of second harmonic generation, on a coupled resonator are compared to theoretical calculations from the proposed model. Copyright © 2010 Elsevier B.V. All rights reserved.

A Procedure to Construct Exact Solutions of Nonlinear Fractional Differential Equations

PubMed Central

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

A procedure to construct exact solutions of nonlinear fractional differential equations.

PubMed

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.

New nonlinear evolution equations from surface theory

NASA Astrophysics Data System (ADS)

Gürses, Metin; Nutku, Yavuz

1981-07-01

We point out that the connection between surfaces in three-dimensional flat space and the inverse scattering problem provides a systematic way for constructing new nonlinear evolution equations. In particular we study the imbedding for Guichard surfaces which gives rise to the Calapso-Guichard equations generalizing the sine-Gordon (SG) equation. Further, we investigate the geometry of surfaces and their imbedding which results in the Korteweg-deVries (KdV) equation. Then by constructing a family of applicable surfaces we obtain a generalization of the KdV equation to a compressible fluid.

Lattice Boltzmann model for high-order nonlinear partial differential equations

NASA Astrophysics Data System (ADS)

Chai, Zhenhua; He, Nanzhong; Guo, Zhaoli; Shi, Baochang

2018-01-01

In this paper, a general lattice Boltzmann (LB) model is proposed for the high-order nonlinear partial differential equation with the form ∂tϕ +∑k=1mαk∂xkΠk(ϕ ) =0 (1 ≤k ≤m ≤6 ), αk are constant coefficients, Πk(ϕ ) are some known differential functions of ϕ . As some special cases of the high-order nonlinear partial differential equation, the classical (m)KdV equation, KdV-Burgers equation, K (n ,n ) -Burgers equation, Kuramoto-Sivashinsky equation, and Kawahara equation can be solved by the present LB model. Compared to the available LB models, the most distinct characteristic of the present model is to introduce some suitable auxiliary moments such that the correct moments of equilibrium distribution function can be achieved. In addition, we also conducted a detailed Chapman-Enskog analysis, and found that the high-order nonlinear partial differential equation can be correctly recovered from the proposed LB model. Finally, a large number of simulations are performed, and it is found that the numerical results agree with the analytical solutions, and usually the present model is also more accurate than the existing LB models [H. Lai and C. Ma, Sci. China Ser. G 52, 1053 (2009), 10.1007/s11433-009-0149-3; H. Lai and C. Ma, Phys. A (Amsterdam) 388, 1405 (2009), 10.1016/j.physa.2009.01.005] for high-order nonlinear partial differential equations.

Lattice Boltzmann model for high-order nonlinear partial differential equations.

PubMed

Chai, Zhenhua; He, Nanzhong; Guo, Zhaoli; Shi, Baochang

2018-01-01

In this paper, a general lattice Boltzmann (LB) model is proposed for the high-order nonlinear partial differential equation with the form ∂_{t}ϕ+∑_{k=1}^{m}α_{k}∂_{x}^{k}Π_{k}(ϕ)=0 (1≤k≤m≤6), α_{k} are constant coefficients, Π_{k}(ϕ) are some known differential functions of ϕ. As some special cases of the high-order nonlinear partial differential equation, the classical (m)KdV equation, KdV-Burgers equation, K(n,n)-Burgers equation, Kuramoto-Sivashinsky equation, and Kawahara equation can be solved by the present LB model. Compared to the available LB models, the most distinct characteristic of the present model is to introduce some suitable auxiliary moments such that the correct moments of equilibrium distribution function can be achieved. In addition, we also conducted a detailed Chapman-Enskog analysis, and found that the high-order nonlinear partial differential equation can be correctly recovered from the proposed LB model. Finally, a large number of simulations are performed, and it is found that the numerical results agree with the analytical solutions, and usually the present model is also more accurate than the existing LB models [H. Lai and C. Ma, Sci. China Ser. G 52, 1053 (2009)1672-179910.1007/s11433-009-0149-3; H. Lai and C. Ma, Phys. A (Amsterdam) 388, 1405 (2009)PHYADX0378-437110.1016/j.physa.2009.01.005] for high-order nonlinear partial differential equations.

Optimal Variational Asymptotic Method for Nonlinear Fractional Partial Differential Equations.

PubMed

Baranwal, Vipul K; Pandey, Ram K; Singh, Om P

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 γ 0, γ 1, γ 2,… and auxiliary functions H 0(x), H 1(x), H 2(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.

Hierarchically partitioned nonlinear equation solvers

NASA Technical Reports Server (NTRS)

Padovan, Joseph

1987-01-01

By partitioning solution space into a number of subspaces, a new multiply constrained partitioned Newton-Raphson nonlinear equation solver is developed. Specifically, for a given iteration, each of the various separate partitions are individually and simultaneously controlled. Due to the generality of the scheme, a hierarchy of partition levels can be employed. For finite-element-type applications, this includes the possibility of degree-of-freedom, nodal, elemental, geometric substructural, material and kinematically nonlinear group controls. It is noted that such partitioning can be continuously updated, depending on solution conditioning. In this context, convergence is ascertained at the individual partition level.

Evolutionary algorithm based heuristic scheme for nonlinear heat transfer equations.

PubMed

Ullah, Azmat; Malik, Suheel Abdullah; Alimgeer, Khurram Saleem

2018-01-01

In this paper, a hybrid heuristic scheme based on two different basis functions i.e. Log Sigmoid and Bernstein Polynomial with unknown parameters is used for solving the nonlinear heat transfer equations efficiently. The proposed technique transforms the given nonlinear ordinary differential equation into an equivalent global error minimization problem. Trial solution for the given nonlinear differential equation is formulated using a fitness function with unknown parameters. The proposed hybrid scheme of Genetic Algorithm (GA) with Interior Point Algorithm (IPA) is opted to solve the minimization problem and to achieve the optimal values of unknown parameters. The effectiveness of the proposed scheme is validated by solving nonlinear heat transfer equations. The results obtained by the proposed scheme are compared and found in sharp agreement with both the exact solution and solution obtained by Haar Wavelet-Quasilinearization technique which witnesses the effectiveness and viability of the suggested scheme. Moreover, the statistical analysis is also conducted for investigating the stability and reliability of the presented scheme.

Modified harmonic balance method for the solution of nonlinear jerk equations

NASA Astrophysics Data System (ADS)

Rahman, M. Saifur; Hasan, A. S. M. Z.

2018-03-01

In this paper, a second approximate solution of nonlinear jerk equations (third order differential equation) can be obtained by using modified harmonic balance method. The method is simpler and easier to carry out the solution of nonlinear differential equations due to less number of nonlinear equations are required to solve than the classical harmonic balance method. The results obtained from this method are compared with those obtained from the other existing analytical methods that are available in the literature and the numerical method. The solution shows a good agreement with the numerical solution as well as the analytical methods of the available literature.

Evans functions and bifurcations of nonlinear waves of some nonlinear reaction diffusion equations

NASA Astrophysics Data System (ADS)

Zhang, Linghai

2017-10-01

The main purposes of this paper are to accomplish the existence, stability, instability and bifurcation of the nonlinear waves of the nonlinear system of reaction diffusion equations ut =uxx + α [ βH (u - θ) - u ] - w, wt = ε (u - γw) and to establish the existence, stability, instability and bifurcation of the nonlinear waves of the nonlinear scalar reaction diffusion equation ut =uxx + α [ βH (u - θ) - u ], under different conditions on the model constants. To establish the bifurcation for the system, we will study the existence and instability of a standing pulse solution if 0 < 2 (1 + αγ) θ < αβγ; the existence and stability of two standing wave fronts if 2 (1 + αγ) θ = αβγ and γ2 ε > 1; the existence and instability of two standing wave fronts if 2 (1 + αγ) θ = αβγ and 0 <γ2 ε < 1; the existence and instability of an upside down standing pulse solution if 0 < (1 + αγ) θ < αβγ < 2 (1 + αγ) θ. To establish the bifurcation for the scalar equation, we will study the existence and stability of a traveling wave front as well as the existence and instability of a standing pulse solution if 0 < 2 θ < β; the existence and stability of two standing wave fronts if 2 θ = β; the existence and stability of a traveling wave front as well as the existence and instability of an upside down standing pulse solution if 0 < θ < β < 2 θ. By the way, we will also study the existence and stability of a traveling wave back of the nonlinear scalar reaction diffusion equation ut =uxx + α [ βH (u - θ) - u ] -w0, where w0 = α (β - 2 θ) > 0 is a positive constant, if 0 < 2 θ < β. To achieve the main goals, we will make complete use of the special structures of the model equations and we will construct Evans functions and apply them to study the eigenvalues and eigenfunctions of several eigenvalue problems associated with several linear differential operators. It turns out that a complex number λ0 is an eigenvalue of the linear

On implicit abstract neutral nonlinear differential equations

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

Hernández, Eduardo, E-mail: lalohm@ffclrp.usp.br; O’Regan, Donal, E-mail: donal.oregan@nuigalway.ie

2016-04-15

In this paper we continue our developments in Hernández and O’Regan (J Funct Anal 261:3457–3481, 2011) on the existence of solutions for abstract neutral differential equations. In particular we extend the results in Hernández and O’Regan (J Funct Anal 261:3457–3481, 2011) for the case of implicit nonlinear neutral equations and we focus on applications to partial “nonlinear” neutral differential equations. Some applications involving partial neutral differential equations are presented.

Transport equations for subdiffusion with nonlinear particle interaction.

PubMed

Straka, P; Fedotov, S

2015-02-07

We show how the nonlinear interaction effects 'volume filling' and 'adhesion' can be incorporated into the fractional subdiffusive transport of cells and individual organisms. To this end, we use microscopic random walk models with anomalous trapping and systematically derive generic non-Markovian and nonlinear governing equations for the mean concentrations of the subdiffusive cells or organisms. We uncover an interesting interaction between the nonlinearities and the non-Markovian nature of the transport. In the subdiffusive case, this interaction manifests itself in a nontrivial combination of nonlinear terms with fractional derivatives. In the long time limit, however, these equations simplify to a form without fractional operators. This provides an easy method for the study of aggregation phenomena. In particular, this enables us to show that volume filling can prevent "anomalous aggregation," which occurs in subdiffusive systems with a spatially varying anomalous exponent. Copyright © 2014 Elsevier Ltd. All rights reserved.

A numerical and experimental study on the nonlinear evolution of long-crested irregular waves

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

Goullet, Arnaud; Choi, Wooyoung; Division of Ocean Systems Engineering, Korea Advanced Institute of Science and Technology, Daejeon 305-701

2011-01-15

The spatial evolution of nonlinear long-crested irregular waves characterized by the JONSWAP spectrum is studied numerically using a nonlinear wave model based on a pseudospectral (PS) method and the modified nonlinear Schroedinger (MNLS) equation. In addition, new laboratory experiments with two different spectral bandwidths are carried out and a number of wave probe measurements are made to validate these two wave models. Strongly nonlinear wave groups are observed experimentally and their propagation and interaction are studied in detail. For the comparison with experimental measurements, the two models need to be initialized with care and the initialization procedures are described. Themore » MNLS equation is found to approximate reasonably well for the wave fields with a relatively smaller Benjamin-Feir index, but the phase error increases as the propagation distance increases. The PS model with different orders of nonlinear approximation is solved numerically, and it is shown that the fifth-order model agrees well with our measurements prior to wave breaking for both spectral bandwidths.« less

Current interactions from the one-form sector of nonlinear higher-spin equations

NASA Astrophysics Data System (ADS)

Gelfond, O. A.; Vasiliev, M. A.

2018-06-01

The form of higher-spin current interactions in the sector of one-forms is derived from the nonlinear higher-spin equations in AdS4. Quadratic corrections to higher-spin equations are shown to be independent of the phase of the parameter η = exp ⁡ iφ in the full nonlinear higher-spin equations. The current deformation resulting from the nonlinear higher-spin equations is represented in the canonical form with the minimal number of space-time derivatives. The non-zero spin-dependent coupling constants of the resulting currents are determined in terms of the higher-spin coupling constant η η bar . Our results confirm the conjecture that (anti-)self-dual nonlinear higher-spin equations result from the full system at (η = 0) η bar = 0.

Equating TIMSS Mathematics Subtests with Nonlinear Equating Methods Using NEAT Design: Circle-Arc Equating Approaches

ERIC Educational Resources Information Center

Ozdemir, Burhanettin

2017-01-01

The purpose of this study is to equate Trends in International Mathematics and Science Study (TIMSS) mathematics subtest scores obtained from TIMSS 2011 to scores obtained from TIMSS 2007 form with different nonlinear observed score equating methods under Non-Equivalent Anchor Test (NEAT) design where common items are used to link two or more test…

Nonlinear subdiffusive fractional equations and the aggregation phenomenon.

PubMed

Fedotov, Sergei

2013-09-01

In this article we address the problem of the nonlinear interaction of subdiffusive particles. We introduce the random walk model in which statistical characteristics of a random walker such as escape rate and jump distribution depend on the mean density of particles. We derive a set of nonlinear subdiffusive fractional master equations and consider their diffusion approximations. We show that these equations describe the transition from an intermediate subdiffusive regime to asymptotically normal advection-diffusion transport regime. This transition is governed by nonlinear tempering parameter that generalizes the standard linear tempering. We illustrate the general results through the use of the examples from cell and population biology. We find that a nonuniform anomalous exponent has a strong influence on the aggregation phenomenon.

Finding all solutions of nonlinear equations using the dual simplex method

NASA Astrophysics Data System (ADS)

Yamamura, Kiyotaka; Fujioka, Tsuyoshi

2003-03-01

Recently, an efficient algorithm has been proposed for finding all solutions of systems of nonlinear equations using linear programming. This algorithm is based on a simple test (termed the LP test) for nonexistence of a solution to a system of nonlinear equations using the dual simplex method. In this letter, an improved version of the LP test algorithm is proposed. By numerical examples, it is shown that the proposed algorithm could find all solutions of a system of 300 nonlinear equations in practical computation time.

Detection the nonlinear ultrasonic signals based on modified Duffing equations

NASA Astrophysics Data System (ADS)

Zhang, Yuhua; Mao, Hanling; Mao, Hanying; Huang, Zhenfeng

The nonlinear ultrasonic signals, like second harmonic generation (SHG) signals, could reflect the nonlinearity of material induced by fatigue damage in nonlinear ultrasonic technique which are weak nonlinear signals and usually submerged by strong background noise. In this paper the modified Duffing equations are applied to detect the SHG signals relating to the fatigue damage of material. Due to the Duffing equation could only detect the signal with specific frequency and initial phase, firstly the frequency transformation is carried on the Duffing equation which could detect the signal with any frequency. Then the influence of initial phases of to-be-detected signal and reference signal on the detection result is studied in detail, four modified Duffing equations are proposed to detect actual engineering signals with any initial phase. The relationship between the response amplitude and the total driving force is applied to estimate the amplitude of weak periodic signal. The detection results show the modified Duffing equations could effectively detect the second harmonic in SHG signals. When the SHG signals include strong background noise, the noise doesn't change the motion state of Duffing equation and the second harmonic signal could be detected until the SNR of noisy SHG signals are -26.3, yet the frequency spectrum method could only identify when the SNR is greater than 0.5. When estimation the amplitude of second harmonic signal, the estimation error of Duffing equation is obviously less than the frequency spectrum analysis method under the same noise level, which illustrates the Duffing equation has the noise immune capacity. The presence of the second harmonic signal in nonlinear ultrasonic experiments could provide an insight about the early fatigue damage of engineering components.

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…

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

Additive nonlinear biomass equations: A likelihood-based approach

Treesearch

David L. R. Affleck; Ulises Dieguez-Aranda

2016-01-01

Since Parresol’s (Can. J. For. Res. 31:865-878, 2001) seminal article on the topic, it has become standard to develop nonlinear tree biomass equations to ensure compatibility among total and component predictions and to fit these equations using multistep generalized least-squares methods. In particular, many studies have specified equations for total tree...

A note on improved F-expansion method combined with Riccati equation applied to nonlinear evolution equations.

PubMed

Islam, Md Shafiqul; Khan, Kamruzzaman; Akbar, M Ali; Mastroberardino, Antonio

2014-10-01

The purpose of this article is to present an analytical method, namely the improved F-expansion method combined with the Riccati equation, for finding exact solutions of nonlinear evolution equations. The present method is capable of calculating all branches of solutions simultaneously, even if multiple solutions are very close and thus difficult to distinguish with numerical techniques. To verify the computational efficiency, we consider the modified Benjamin-Bona-Mahony equation and the modified Korteweg-de Vries equation. Our results reveal that the method is a very effective and straightforward way of formulating the exact travelling wave solutions of nonlinear wave equations arising in mathematical physics and engineering.

A note on improved F-expansion method combined with Riccati equation applied to nonlinear evolution equations

PubMed Central

Islam, Md. Shafiqul; Khan, Kamruzzaman; Akbar, M. Ali; Mastroberardino, Antonio

2014-01-01

The purpose of this article is to present an analytical method, namely the improved F-expansion method combined with the Riccati equation, for finding exact solutions of nonlinear evolution equations. The present method is capable of calculating all branches of solutions simultaneously, even if multiple solutions are very close and thus difficult to distinguish with numerical techniques. To verify the computational efficiency, we consider the modified Benjamin–Bona–Mahony equation and the modified Korteweg-de Vries equation. Our results reveal that the method is a very effective and straightforward way of formulating the exact travelling wave solutions of nonlinear wave equations arising in mathematical physics and engineering. PMID:26064530

Traveling wave solutions and conservation laws for nonlinear evolution equation

NASA Astrophysics Data System (ADS)

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

2018-02-01

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

A family of nonlinear Schrödinger equations admitting q-plane wave solutions

NASA Astrophysics Data System (ADS)

Nobre, F. D.; Plastino, A. R.

2017-08-01

Nonlinear Schrödinger equations with power-law nonlinearities have attracted considerable attention recently. Two previous proposals for these types of equations, corresponding respectively to the Gross-Pitaievsky equation and to the one associated with nonextensive statistical mechanics, are here unified into a single, parameterized family of nonlinear Schrödinger equations. Power-law nonlinear terms characterized by exponents depending on a real index q, typical of nonextensive statistical mechanics, are considered in such a way that the Gross-Pitaievsky equation is recovered in the limit q → 1. A classical field theory shows that, due to these nonlinearities, an extra field Φ (x → , t) (besides the usual one Ψ (x → , t)) must be introduced for consistency. The new field can be identified with Ψ* (x → , t) only when q → 1. For q ≠ 1 one has a pair of coupled nonlinear wave equations governing the joint evolution of the complex valued fields Ψ (x → , t) and Φ (x → , t). These equations reduce to the usual pair of complex-conjugate ones only in the q → 1 limit. Interestingly, the nonlinear equations obeyed by Ψ (x → , t) and Φ (x → , t) exhibit a common, soliton-like, traveling solution, which is expressible in terms of the q-exponential function that naturally emerges within nonextensive statistical mechanics.

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.

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.

Explicit expressions for meromorphic solutions of autonomous nonlinear ordinary differential equations

NASA Astrophysics Data System (ADS)

Demina, Maria V.; Kudryashov, Nikolay A.

2011-03-01

Meromorphic solutions of autonomous nonlinear ordinary differential equations are studied. An algorithm for constructing meromorphic solutions in explicit form is presented. General expressions for meromorphic solutions (including rational, periodic, elliptic) are found for a wide class of autonomous nonlinear ordinary differential equations.

Hidden Statistics of Schroedinger Equation

NASA Technical Reports Server (NTRS)

Zak, Michail

2011-01-01

Work was carried out in determination of the mathematical origin of randomness in quantum mechanics and creating a hidden statistics of Schr dinger equation; i.e., to expose the transitional stochastic process as a "bridge" to the quantum world. The governing equations of hidden statistics would preserve such properties of quantum physics as superposition, entanglement, and direct-product decomposability while allowing one to measure its state variables using classical methods.

Entire solutions of nonlinear differential-difference equations.

PubMed

Li, Cuiping; Lü, Feng; Xu, Junfeng

2016-01-01

In this paper, we describe the properties of entire solutions of a nonlinear differential-difference equation and a Fermat type equation, and improve several previous theorems greatly. In addition, we also deduce a uniqueness result for an entire function f(z) that shares a set with its shift [Formula: see text], which is a generalization of a result of Liu.

Canonical equations of Hamilton for the nonlinear Schrödinger equation

NASA Astrophysics Data System (ADS)

Liang, Guo; Guo, Qi; Ren, Zhanmei

2015-09-01

We define two different systems of mathematical physics: the second order differential system (SODS) and the first order differential system (FODS). The Newton's second law of motion and the nonlinear Schrödinger equation (NLSE) are the exemplary SODS and FODS, respectively. We obtain a new kind of canonical equations of Hamilton (CEH), which exhibit some kind of symmetry in form and are formally different from the conventional CEH without symmetry [H. Goldstein, C. Poole, J. Safko, Classical Mechanics, third ed., Addison- Wesley, 2001]. We also prove that the number of the CEHs is equal to the number of the generalized coordinates for the FODS, but twice the number of the generalized coordinates for the SODS. We show that the FODS can only be expressed by the new CEH, but not introduced by the conventional CEH, while the SODS can be done by both the new and the conventional CEHs. As an example, we prove that the nonlinear Schrödinger equation can be expressed with the new CEH in a consistent way.

Nonlinear field equations for aligning self-propelled rods.

PubMed

Peshkov, Anton; Aranson, Igor S; Bertin, Eric; Chaté, Hugues; Ginelli, Francesco

2012-12-28

We derive a set of minimal and well-behaved nonlinear field equations describing the collective properties of self-propelled rods from a simple microscopic starting point, the Vicsek model with nematic alignment. Analysis of their linear and nonlinear dynamics shows good agreement with the original microscopic model. In particular, we derive an explicit expression for density-segregated, banded solutions, allowing us to develop a more complete analytic picture of the problem at the nonlinear level.

Quantum theory of rotational isomerism and Hill equation

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

Ugulava, A.; Toklikishvili, Z.; Chkhaidze, S.

2012-06-15

The process of rotational isomerism of linear triatomic molecules is described by the potential with two different-depth minima and one barrier between them. The corresponding quantum-mechanical equation is represented in the form that is a special case of the Hill equation. It is shown that the Hill-Schroedinger equation has a Klein's quadratic group symmetry which, in its turn, contains three invariant subgroups. The presence of these subgroups makes it possible to create a picture of energy spectrum which depends on a parameter and has many merging and branch points. The parameter-dependent energy spectrum of the Hill-Schroedinger equation, like Mathieu-characteristics, containsmore » branch points from the left and from the right of the demarcation line. However, compared to the Mathieu-characteristics, in the Hill-Schroedinger equation spectrum the 'right' points are moved away even further for some distance that is the bigger, the bigger is the less deep well. The asymptotic wave functions of the Hill-Schroedinger equation for the energy values near the potential minimum contain two isolated sharp peaks indicating a possibility of the presence of two stable isomers. At high energy values near the potential maximum, the height of two peaks decreases, and between them there appear chaotic oscillations. This form of the wave functions corresponds to the process of isomerization.« less

Bayesian Analysis of Structural Equation Models with Nonlinear Covariates and Latent Variables

ERIC Educational Resources Information Center

Song, Xin-Yuan; Lee, Sik-Yum

2006-01-01

In this article, we formulate a nonlinear structural equation model (SEM) that can accommodate covariates in the measurement equation and nonlinear terms of covariates and exogenous latent variables in the structural equation. The covariates can come from continuous or discrete distributions. A Bayesian approach is developed to analyze the…

Exact multisoliton solutions of general nonlinear Schrödinger equation with derivative.

PubMed

Li, Qi; Duan, Qiu-yuan; Zhang, Jian-bing

2014-01-01

Multisoliton solutions are derived for a general nonlinear Schrödinger equation with derivative by using Hirota's approach. The dynamics of one-soliton solution and two-soliton interactions are also illustrated. The considered equation can reduce to nonlinear Schrödinger equation with derivative as well as the solutions.

Chirped femtosecond pulses in the higher-order nonlinear Schrödinger equation with non-Kerr nonlinear terms and cubic-quintic-septic nonlinearities

NASA Astrophysics Data System (ADS)

Triki, Houria; Biswas, Anjan; Milović, Daniela; Belić, Milivoj

2016-05-01

We consider a high-order nonlinear Schrödinger equation with competing cubic-quintic-septic nonlinearities, non-Kerr quintic nonlinearity, self-steepening, and self-frequency shift. The model describes the propagation of ultrashort (femtosecond) optical pulses in highly nonlinear optical fibers. A new ansatz is adopted to obtain nonlinear chirp associated with the propagating femtosecond soliton pulses. It is shown that the resultant elliptic equation of the problem is of high order, contains several new terms and is more general than the earlier reported results, thus providing a systematic way to find exact chirped soliton solutions of the septic model. Novel soliton solutions, including chirped bright, dark, kink and fractional-transform soliton solutions are obtained for special choices of parameters. Furthermore, we present the parameter domains in which these optical solitons exist. The nonlinear chirp associated with each of the solitonic solutions is also determined. It is shown that the chirping is proportional to the intensity of the wave and depends on higher-order nonlinearities. Of special interest is the soliton solution of the bright and dark type, determined for the general case when all coefficients in the equation have nonzero values. These results can be useful for possible chirped-soliton-based applications of highly nonlinear optical fiber systems.

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

NASA Technical Reports Server (NTRS)

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

1994-01-01

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

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

NASA Technical Reports Server (NTRS)

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

1995-01-01

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

Approximating a nonlinear advanced-delayed equation from acoustics

NASA Astrophysics Data System (ADS)

Teodoro, M. Filomena

2016-10-01

We approximate the solution of a particular non-linear mixed type functional differential equation from physiology, the mucosal wave model of the vocal oscillation during phonation. The mathematical equation models a superficial wave propagating through the tissues. The numerical scheme is adapted from the work presented in [1, 2, 3], using homotopy analysis method (HAM) to solve the non linear mixed type equation under study.

A Bivariate Chebyshev Spectral Collocation Quasilinearization Method for Nonlinear Evolution Parabolic Equations

PubMed Central

Motsa, S. S.; Magagula, V. M.; Sibanda, P.

2014-01-01

This paper presents a new method for solving higher order nonlinear evolution partial differential equations (NPDEs). The method combines quasilinearisation, the Chebyshev spectral collocation method, and bivariate Lagrange interpolation. In this paper, we use the method to solve several nonlinear evolution equations, such as the modified KdV-Burgers equation, highly nonlinear modified KdV equation, Fisher's equation, Burgers-Fisher equation, Burgers-Huxley equation, and the Fitzhugh-Nagumo equation. The results are compared with known exact analytical solutions from literature to confirm accuracy, convergence, and effectiveness of the method. There is congruence between the numerical results and the exact solutions to a high order of accuracy. Tables were generated to present the order of accuracy of the method; convergence graphs to verify convergence of the method and error graphs are presented to show the excellent agreement between the results from this study and the known results from literature. PMID:25254252

A bivariate Chebyshev spectral collocation quasilinearization method for nonlinear evolution parabolic equations.

PubMed

Motsa, S S; Magagula, V M; Sibanda, P

2014-01-01

This paper presents a new method for solving higher order nonlinear evolution partial differential equations (NPDEs). The method combines quasilinearisation, the Chebyshev spectral collocation method, and bivariate Lagrange interpolation. In this paper, we use the method to solve several nonlinear evolution equations, such as the modified KdV-Burgers equation, highly nonlinear modified KdV equation, Fisher's equation, Burgers-Fisher equation, Burgers-Huxley equation, and the Fitzhugh-Nagumo equation. The results are compared with known exact analytical solutions from literature to confirm accuracy, convergence, and effectiveness of the method. There is congruence between the numerical results and the exact solutions to a high order of accuracy. Tables were generated to present the order of accuracy of the method; convergence graphs to verify convergence of the method and error graphs are presented to show the excellent agreement between the results from this study and the known results from literature.

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

NASA Technical Reports Server (NTRS)

Mickens, R. E.

1985-01-01

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

Explicit integration of Friedmann's equation with nonlinear equations of state

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

Chen, Shouxin; Gibbons, Gary W.; Yang, Yisong, E-mail: chensx@henu.edu.cn, E-mail: gwg1@damtp.cam.ac.uk, E-mail: yisongyang@nyu.edu

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 generalmore » 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.« less

Universal and integrable nonlinear evolution systems of equations in 2+1 dimensions

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

Maccari, A.

1997-08-01

Integrable systems of nonlinear partial differential equations (PDEs) are obtained from integrable equations in 2+1 dimensions, by means of a reduction method of broad applicability based on Fourier expansion and spatio{endash}temporal rescalings, which is asymptotically exact in the limit of weak nonlinearity. The integrability by the spectral transform is explicitly demonstrated, because the corresponding Lax pairs have been derived, applying the same reduction method to the Lax pair of the initial equation. These systems of nonlinear PDEs are likely to be of applicative relevance and have a {open_quotes}universal{close_quotes} character, inasmuch as they may be derived from a very large classmore » of nonlinear evolution equations with a linear dispersive part. {copyright} {ital 1997 American Institute of Physics.}« less

Numerical calculation of nonlinear ultrashort laser pulse propagation in transparent Kerr media

NASA Astrophysics Data System (ADS)

Arnold, Cord L.; Heisterkamp, Alexander; Ertmer, Wolfgang; Lubatschowski, Holger

2005-03-01

In the focal region of tightly focused ultrashort laser pulses, sufficient high intensities to initialize nonlinear ionization processes are easily achieved. Due to these nonlinear ionization processes, mainly multiphoton ionization and cascade ionization, free electrons are generated in the focus resulting in optical breakdown. A model including both nonlinear pulse propagation and plasma generation is used to calculate numerically the interaction of ultrashort pulses with their self-induced plasma in the vicinity of the focus. The model is based on a (3+1)-dimensional nonlinear Schroedinger equation describing the pulse propagation coupled to a system of rate equations covering the generation of free electrons. It is applicable to any transparent Kerr medium, whose linear and nonlinear optical parameters are known. Numerical calculations based on this model are used to understand nonlinear side effects, such as streak formation, occurring in addition to optical breakdown during short pulse refractive eye surgeries like fs-LASIK. Since the optical parameters of water are a good first-order approximation to those of corneal tissue, water is used as model substance. The free electron density distribution induced by focused ultrashort pulses as well as the pulses spatio-temporal behavior are studied in the low-power regime around the critical power for self-focusing.

Nonlinear acoustic wave equations with fractional loss operators.

PubMed

Prieur, Fabrice; Holm, Sverre

2011-09-01

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

A Jacobi collocation approximation for nonlinear coupled viscous Burgers' equation

NASA Astrophysics Data System (ADS)

Doha, Eid H.; Bhrawy, Ali H.; Abdelkawy, Mohamed A.; Hafez, Ramy M.

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.

Curl forces and the nonlinear Fokker-Planck equation.

PubMed

Wedemann, R S; Plastino, A R; Tsallis, C

2016-12-01

Nonlinear Fokker-Planck equations endowed with curl drift forces are investigated. The conditions under which these evolution equations admit stationary solutions, which are q exponentials of an appropriate potential function, are determined. It is proved that when these stationary solutions exist, the nonlinear Fokker-Planck equations satisfy an H theorem in terms of a free-energy-like quantity involving the S_{q} entropy. A particular two-dimensional model admitting analytical, time-dependent q-Gaussian solutions is discussed in detail. This model describes a system of particles with short-range interactions, performing overdamped motion under drag effects due to a rotating resisting medium. It is related to models that have been recently applied to the study of type-II superconductors. The relevance of the present developments to the study of complex systems in physics, astronomy, and biology is discussed.

Maximum Likelihood Analysis of Nonlinear Structural Equation Models with Dichotomous Variables

ERIC Educational Resources Information Center

Song, Xin-Yuan; Lee, Sik-Yum

2005-01-01

In this article, a maximum likelihood approach is developed to analyze structural equation models with dichotomous variables that are common in behavioral, psychological and social research. To assess nonlinear causal effects among the latent variables, the structural equation in the model is defined by a nonlinear function. The basic idea of the…

Applications of Nonlinear Control Using the State-Dependent Riccati Equation.

DTIC Science & Technology

1995-12-01

method, and do not address noise rejection or robustness issues. xi Applications of Nonlinear Control Using the State-Dependent Riccati Equation I...construct a stabilizing nonlinear feedback controller. This method will be referred to as nonlinear quadratic regulation (NQR). The original intention...involves nding a state-dependent coe- cient (SDC) linear structure for which a stabilizing nonlinear feedback controller can be constructed. The

From nonlinear Schrödinger hierarchy to some (2+1)-dimensional nonlinear pseudodifferential equations

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.

Semiclassical regularization of Vlasov equations and wavepackets for nonlinear Schrödinger equations

NASA Astrophysics Data System (ADS)

Athanassoulis, Agissilaos

2018-03-01

We consider the semiclassical limit of nonlinear Schrödinger equations with initial data that are well localized in both position and momentum (non-parametric wavepackets). We recover the Wigner measure (WM) of the problem, a macroscopic phase-space density which controls the propagation of the physical observables such as mass, energy and momentum. WMs have been used to create effective models for wave propagation in: random media, quantum molecular dynamics, mean field limits, and the propagation of electrons in graphene. In nonlinear settings, the Vlasov-type equations obtained for the WM are often ill-posed on the physically interesting spaces of initial data. In this paper we are able to select the measure-valued solution of the 1  +  1 dimensional Vlasov-Poisson equation which correctly captures the semiclassical limit, thus finally resolving the non-uniqueness in the seminal result of Zhang et al (2012 Comm. Pure Appl. Math. 55 582-632). The same approach is also applied to the Vlasov-Dirac-Benney equation with small wavepacket initial data, extending several known results.

Dynamical systems theory for nonlinear evolution equations.

PubMed

Choudhuri, Amitava; Talukdar, B; Das, Umapada

2010-09-01

We observe that the fully nonlinear evolution equations of Rosenau and Hymann, often abbreviated as K(n,m) equations, can be reduced to Hamiltonian form only on a zero-energy hypersurface belonging to some potential function associated with the equations. We treat the resulting Hamiltonian equations by the dynamical systems theory and present a phase-space analysis of their stable points. The results of our study demonstrate that the equations can, in general, support both compacton and soliton solutions. For the K(2,2) and K(3,3) cases one type of solutions can be obtained from the other by continuously varying a parameter of the equations. This is not true for the K(3,2) equation for which the parameter can take only negative values. The K(2,3) equation does not have any stable point and, in the language of mechanics, represents a particle moving with constant acceleration.

How Accurately Does the Free Complement Wave Function of a Helium Atom Satisfy the Schroedinger Equation?

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

Nakashima, Hiroyuki; Nakatsuji, Hiroshi

2008-12-12

The local energy defined by H{psi}/{psi} must be equal to the exact energy E at any coordinate of an atom or molecule, as long as the {psi} under consideration is exact. The discrepancy from E of this quantity is a stringent test of the accuracy of the calculated wave function. The H-square error for a normalized {psi}, defined by {sigma}{sup 2}{identical_to}<{psi}|(H-E){sup 2}|{psi}>, is also a severe test of the accuracy. Using these quantities, we have examined the accuracy of our wave function of a helium atom calculated using the free complement method that was developed to solve the Schroedinger equation.more » Together with the variational upper bound, the lower bound of the exact energy calculated using a modified Temple's formula ensured the definitely correct value of the helium fixed-nucleus ground state energy to be -2.903 724 377 034 119 598 311 159 245 194 4 a.u., which is correct to 32 digits.« less

An ansatz for solving nonlinear partial differential equations in mathematical physics.

PubMed

Akbar, M Ali; Ali, Norhashidah Hj Mohd

2016-01-01

In this article, we introduce an ansatz involving exact traveling wave solutions to nonlinear partial differential equations. To obtain wave solutions using direct method, the choice of an appropriate ansatz is of great importance. We apply this ansatz to examine new and further general traveling wave solutions to the (1+1)-dimensional modified Benjamin-Bona-Mahony equation. Abundant traveling wave solutions are derived including solitons, singular solitons, periodic solutions and general solitary wave solutions. The solutions emphasize the nobility of this ansatz in providing distinct solutions to various tangible phenomena in nonlinear science and engineering. The ansatz could be more efficient tool to deal with higher dimensional nonlinear evolution equations which frequently arise in many real world physical problems.

Study of travelling wave solutions for some special-type nonlinear evolution equations

NASA Astrophysics Data System (ADS)

Song, Junquan; Hu, Lan; Shen, Shoufeng; Ma, Wen-Xiu

2018-07-01

The tanh-function expansion method has been improved and used to construct travelling wave solutions of the form U={\\sum }j=0n{a}j{\\tanh }jξ for some special-type nonlinear evolution equations, which have a variety of physical applications. The positive integer n can be determined by balancing the highest order linear term with the nonlinear term in the evolution equations. We improve the tanh-function expansion method with n = 0 by introducing a new transform U=-W\\prime (ξ )/{W}2. A nonlinear wave equation with source terms, and mKdV-type equations, are considered in order to show the effectiveness of the improved scheme. We also propose the tanh-function expansion method of implicit function form, and apply it to a Harry Dym-type equation as an example.

Optical solitons to the resonance nonlinear Schrödinger equation by Sine-Gordon equation method

NASA Astrophysics Data System (ADS)

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

2018-01-01

In this paper, we examined the optical solitons to the resonant nonlinear Schrödinger equation (R-NLSE) which describes the propagation of solitons through optical fibers. Three types of nonlinear media fibers are studied. They are; quadratic-cubic law, Kerr law and parabolic law. Dark, bright, dark-bright or combined optical and singular soliton solutions are derived using the sine-Gordon equation method (SGEM). The constraint conditions that naturally fall out of the solution structure which guarantee the existence of these solitons are also reported.

Parabolic equation for nonlinear acoustic wave propagation in inhomogeneous moving media

NASA Astrophysics Data System (ADS)

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

2006-12-01

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

Traveling wavefront solutions to nonlinear reaction-diffusion-convection equations

NASA Astrophysics Data System (ADS)

Indekeu, Joseph O.; Smets, Ruben

2017-08-01

Physically motivated modified Fisher equations are studied in which nonlinear convection and nonlinear diffusion is allowed for besides the usual growth and spread of a population. It is pointed out that in a large variety of cases separable functions in the form of exponentially decaying sharp wavefronts solve the differential equation exactly provided a co-moving point source or sink is active at the wavefront. The velocity dispersion and front steepness may differ from those of some previously studied exact smooth traveling wave solutions. For an extension of the reaction-diffusion-convection equation, featuring a memory effect in the form of a maturity delay for growth and spread, also smooth exact wavefront solutions are obtained. The stability of the solutions is verified analytically and numerically.

State-Dependent Riccati Equation Regulation of Systems with State and Control Nonlinearities

NASA Technical Reports Server (NTRS)

Beeler, Scott C.; Cox, David E. (Technical Monitor)

2004-01-01

The state-dependent Riccati equations (SDRE) is the basis of a technique for suboptimal feedback control of a nonlinear quadratic regulator (NQR) problem. It is an extension of the Riccati equation used for feedback control of linear problems, with the addition of nonlinearities in the state dynamics of the system resulting in a state-dependent gain matrix as the solution of the equation. In this paper several variations on the SDRE-based method will be considered for the feedback control problem with control nonlinearities. The control nonlinearities may result in complications in the numerical implementation of the control, which the different versions of the SDRE method must try to overcome. The control methods will be applied to three test problems and their resulting performance analyzed.

Study of solution procedures for nonlinear structural equations

NASA Technical Reports Server (NTRS)

Young, C. T., II; Jones, R. F., Jr.

1980-01-01

A method for the redution of the cost of solution of large nonlinear structural equations was developed. Verification was made using the MARC-STRUC structure finite element program with test cases involving single and multiple degrees of freedom for static geometric nonlinearities. The method developed was designed to exist within the envelope of accuracy and convergence characteristic of the particular finite element methodology used.

On symmetries, conservation laws and exact solutions of the nonlinear Schrödinger-Hirota equation

NASA Astrophysics Data System (ADS)

Akbulut, Arzu; Taşcan, Filiz

2018-04-01

In this paper, conservation laws and exact solution are found for nonlinear Schrödinger-Hirota equation. Conservation theorem is used for finding conservation laws. We get modified conservation laws for given equation. Modified simple equation method is used to obtain the exact solutions of the nonlinear Schrödinger-Hirota equation. It is shown that the suggested method provides a powerful mathematical instrument for solving nonlinear equations in mathematical physics and engineering.

Exact solutions to three-dimensional generalized nonlinear Schrödinger equations with varying potential and nonlinearities.

PubMed

Yan, Zhenya; Konotop, V V

2009-09-01

It is shown that using the similarity transformations, a set of three-dimensional p-q nonlinear Schrödinger (NLS) equations with inhomogeneous coefficients can be reduced to one-dimensional stationary NLS equation with constant or varying coefficients, thus allowing for obtaining exact localized and periodic wave solutions. In the suggested reduction the original coordinates in the (1+3) space are mapped into a set of one-parametric coordinate surfaces, whose parameter plays the role of the coordinate of the one-dimensional equation. We describe the algorithm of finding solutions and concentrate on power (linear and nonlinear) potentials presenting a number of case examples. Generalizations of the method are also discussed.

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.

1/f Noise from nonlinear stochastic differential equations.

PubMed

Ruseckas, J; Kaulakys, B

2010-03-01

We consider a class of nonlinear stochastic differential equations, giving the power-law behavior of the power spectral density in any desirably wide range of frequency. Such equations were obtained starting from the point process models of 1/fbeta noise. In this article the power-law behavior of spectrum is derived directly from the stochastic differential equations, without using the point process models. The analysis reveals that the power spectrum may be represented as a sum of the Lorentzian spectra. Such a derivation provides additional justification of equations, expands the class of equations generating 1/fbeta noise, and provides further insights into the origin of 1/fbeta noise.

Tensor-GMRES method for large sparse systems of nonlinear equations

NASA Technical Reports Server (NTRS)

Feng, Dan; Pulliam, Thomas H.

1994-01-01

This paper introduces a tensor-Krylov method, the tensor-GMRES method, for large sparse systems of nonlinear equations. This method is a coupling of tensor model formation and solution techniques for nonlinear equations with Krylov subspace projection techniques for unsymmetric systems of linear equations. Traditional tensor methods for nonlinear equations are based on a quadratic model of the nonlinear function, a standard linear model augmented by a simple second order term. These methods are shown to be significantly more efficient than standard methods both on nonsingular problems and on problems where the Jacobian matrix at the solution is singular. A major disadvantage of the traditional tensor methods is that the solution of the tensor model requires the factorization of the Jacobian matrix, which may not be suitable for problems where the Jacobian matrix is large and has a 'bad' sparsity structure for an efficient factorization. We overcome this difficulty by forming and solving the tensor model using an extension of a Newton-GMRES scheme. Like traditional tensor methods, we show that the new tensor method has significant computational advantages over the analogous Newton counterpart. Consistent with Krylov subspace based methods, the new tensor method does not depend on the factorization of the Jacobian matrix. As a matter of fact, the Jacobian matrix is never needed explicitly.

Soliton Resolution for the Derivative Nonlinear Schrödinger Equation

NASA Astrophysics Data System (ADS)

Jenkins, Robert; Liu, Jiaqi; Perry, Peter; Sulem, Catherine

2018-05-01

We study the derivative nonlinear Schrödinger equation for generic initial data in a weighted Sobolev space that can support bright solitons (but exclude spectral singularities). Drawing on previous well-posedness results, we give a full description of the long-time behavior of the solutions in the form of a finite sum of localized solitons and a dispersive component. At leading order and in space-time cones, the solution has the form of a multi-soliton whose parameters are slightly modified from their initial values by soliton-soliton and soliton-radiation interactions. Our analysis provides an explicit expression for the correction dispersive term. We use the nonlinear steepest descent method of Deift and Zhou (Commun Pure Appl Math 56:1029-1077, 2003) revisited by the {\\overline{partial}} -analysis of McLaughlin and Miller (IMRP Int Math Res Pap 48673:1-77, 2006) and Dieng and McLaughlin (Long-time asymptotics for the NLS equation via dbar methods. Preprint, arXiv:0805.2807, 2008), and complemented by the recent work of Borghese et al. (Ann Inst Henri Poincaré Anal Non Linéaire, https://doi.org/10.1016/j.anihpc.2017.08.006, 2017) on soliton resolution for the focusing nonlinear Schrödinger equation. Our results imply that N-soliton solutions of the derivative nonlinear Schrödinger equation are asymptotically stable.

Dissipative behavior of some fully non-linear KdV-type equations

NASA Astrophysics Data System (ADS)

Brenier, Yann; Levy, Doron

2000-03-01

The KdV equation can be considered as a special case of the general equation u t+f(u) x-δg(u xx) x=0, δ>0, where f is non-linear and g is linear, namely f( u)= u2/2 and g( v)= v. As the parameter δ tends to 0, the dispersive behavior of the KdV equation has been throughly investigated (see, e.g., [P.G. Drazin, Solitons, London Math. Soc. Lect. Note Ser. 85, Cambridge University Press, Cambridge, 1983; P.D. Lax, C.D. Levermore, The small dispersion limit of the Korteweg-de Vries equation, III, Commun. Pure Appl. Math. 36 (1983) 809-829; G.B. Whitham, Linear and Nonlinear Waves, Wiley/Interscience, New York, 1974] and the references therein). We show through numerical evidence that a completely different, dissipative behavior occurs when g is non-linear, namely when g is an even concave function such as g( v)=-∣ v∣ or g( v)=- v2. In particular, our numerical results hint that as δ→0 the solutions strongly converge to the unique entropy solution of the formal limit equation, in total contrast with the solutions of the KdV equation.

Exactly Solvable Multidimensional Nonlinear Equations and Inverse Scattering,

DTIC Science & Technology

1986-12-01

time dimension. Here the prototype euQation is 1 the Kadomtsev - Petviashvili (K-P) equation : .0 6u , x , x - )3,:’u ,’ which is the cop,patliil ity...AD-R193 274 EXACTLY SOLVABLE MULTIDIMENSIONAL NONLINEAR EQUATIONS L/1 AND INVERSE SCATTERING(U) CLARKSON UNIV POTSDAM MY A J MBLOUITZ DEC 86 NSOSI4...ecuations by associating thnm with appropriate compatible linear equations , -ne of which is identified as a Scattering prooD,, ne others(s) serves to

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

NASA Astrophysics Data System (ADS)

Liu, Wei

2017-10-01

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

Taming the nonlinearity of the Einstein equation.

PubMed

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.

Square-integrable solutions to a family of nonlinear schrödinger equations from nonlinear quantum theory

NASA Astrophysics Data System (ADS)

Teismann, Holger

2005-10-01

We consider nonlinear Schrödinger equations which have been proposed as fundamental equations of nonlinear quantum theories. The equations are singular in that the wave function ψ appears in the denominator of rational expressions. To avoid the problem of zeros of ψ it is natural to make the ansatz ψ = e ν. This ansatz, however, conflicts with the—physically motivated—requirement that the solutions ψ be square integrable. We show that this conflict can be resolved by considering an unusual function space whose definition involves the derivative ∇ ν of ν. This function space turns out to be dense subset of L2 and the equations can be solved in the L2-sense (as desired) by first solving an evolutionary system for ∇ ν and then transforming back to ψ.

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.

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

NASA Astrophysics Data System (ADS)

Yang, Bo; Chen, Yong

2018-05-01

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

Solving Nonlinear Differential Equations in the Engineering Curriculum

ERIC Educational Resources Information Center

Auslander, David M.

1977-01-01

Described is the Dynamic System Simulation Language (SIM) mini-computer system utilized at the University of California, Los Angeles. It is used by engineering students for solving nonlinear differential equations. (SL)

Initial study of Schroedinger eigenmaps for spectral target detection

NASA Astrophysics Data System (ADS)

Dorado-Munoz, Leidy P.; Messinger, David W.

2016-08-01

Spectral target detection refers to the process of searching for a specific material with a known spectrum over a large area containing materials with different spectral signatures. Traditional target detection methods in hyperspectral imagery (HSI) require assuming the data fit some statistical or geometric models and based on the model, to estimate parameters for defining a hypothesis test, where one class (i.e., target class) is chosen over the other classes (i.e., background class). Nonlinear manifold learning methods such as Laplacian eigenmaps (LE) have extensively shown their potential use in HSI processing, specifically in classification or segmentation. Recently, Schroedinger eigenmaps (SE), which is built upon LE, has been introduced as a semisupervised classification method. In SE, the former Laplacian operator is replaced by the Schroedinger operator. The Schroedinger operator includes by definition, a potential term V that steers the transformation in certain directions improving the separability between classes. In this regard, we propose a methodology for target detection that is not based on the traditional schemes and that does not need the estimation of statistical or geometric parameters. This method is based on SE, where the potential term V is taken into consideration to include the prior knowledge about the target class and use it to steer the transformation in directions where the target location in the new space is known and the separability between target and background is augmented. An initial study of how SE can be used in a target detection scheme for HSI is shown here. In-scene pixel and spectral signature detection approaches are presented. The HSI data used comprise various target panels for testing simultaneous detection of multiple objects with different complexities.

Integrable equations of the infinite nonlinear Schrödinger equation hierarchy with time variable coefficients.

PubMed

Kedziora, D J; Ankiewicz, A; Chowdury, A; Akhmediev, N

2015-10-01

We present an infinite nonlinear Schrödinger equation hierarchy of integrable equations, together with the recurrence relations defining it. To demonstrate integrability, we present the Lax pairs for the whole hierarchy, specify its Darboux transformations and provide several examples of solutions. These resulting wavefunctions are given in exact analytical form. We then show that the Lax pair and Darboux transformation formalisms still apply in this scheme when the coefficients in the hierarchy depend on the propagation variable (e.g., time). This extension thus allows for the construction of complicated solutions within a greatly diversified domain of generalised nonlinear systems.

Dispersive optical soliton solutions for higher order nonlinear Sasa-Satsuma equation in mono mode fibers via new auxiliary equation method

NASA Astrophysics Data System (ADS)

Khater, Mostafa M. A.; Seadawy, Aly R.; Lu, Dianchen

2018-01-01

In this research, we apply new technique for higher order nonlinear Schrödinger equation which is representing the propagation of short light pulses in the monomode optical fibers and the evolution of slowly varying packets of quasi-monochromatic waves in weakly nonlinear media that have dispersion. Nonlinear Schrödinger equation is one of the basic model in fiber optics. We apply new auxiliary equation method for nonlinear Sasa-Satsuma equation to obtain a new optical forms of solitary traveling wave solutions. Exact and solitary traveling wave solutions are obtained in different kinds like trigonometric, hyperbolic, exponential, rational functions, …, etc. These forms of solutions that we represent in this research prove the superiority of our new technique on almost thirteen powerful methods. The main merits of this method over the other methods are that it gives more general solutions with some free parameters.

On solutions of the fifth-order dispersive equations with porous medium type non-linearity

NASA Astrophysics Data System (ADS)

Kocak, Huseyin; Pinar, Zehra

2018-07-01

In this work, we focus on obtaining the exact solutions of the fifth-order semi-linear and non-linear dispersive partial differential equations, which have the second-order diffusion-like (porous-type) non-linearity. The proposed equations were not studied in the literature in the sense of the exact solutions. We reveal solutions of the proposed equations using the classical Riccati equations method. The obtained exact solutions, which can play a key role to simulate non-linear waves in the medium with dispersion and diffusion, are illustrated and discussed in details.

Exact solutions for (1 + 1)-dimensional nonlinear dispersive modified Benjamin-Bona-Mahony equation and coupled Klein-Gordon equations.

PubMed

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

2014-01-01

In this work, recently developed modified simple equation (MSE) method is applied to find exact traveling wave solutions of nonlinear evolution equations (NLEEs). To do so, we consider the (1 + 1)-dimensional nonlinear dispersive modified Benjamin-Bona-Mahony (DMBBM) equation and coupled Klein-Gordon (cKG) equations. Two classes of explicit exact solutions-hyperbolic and trigonometric solutions of the associated equations are characterized with some free parameters. Then these exact solutions correspond to solitary waves for particular values of the parameters. 02.30.Jr; 02.70.Wz; 05.45.Yv; 94.05.Fg.

Nonlinear model of a rotating hub-beams structure: Equations of motion

NASA Astrophysics Data System (ADS)

Warminski, Jerzy

2018-01-01

Dynamics of a rotating structure composed of a rigid hub and flexible beams is presented in the paper. A nonlinear model of a beam takes into account bending, extension and nonlinear curvature. The influence of geometric nonlinearity and nonconstant angular velocity on dynamics of the rotating structure is presented. The exact equations of motion and associated boundary conditions are derived on the basis of the Hamilton's principle. The simplification of the exact nonlinear mathematical model is proposed taking into account the second order approximation. The reduced partial differential equations of motion together with associated boundary conditions can be used to study natural or forced vibrations of a rotating structure considering constant or nonconstant angular speed of a rigid hub and an arbitrary number of flexible blades.

An efficient computational method for the approximate solution of nonlinear Lane-Emden type equations arising in astrophysics

NASA Astrophysics Data System (ADS)

Singh, Harendra

2018-04-01

The key purpose of this article is to introduce an efficient computational method for the approximate solution of the homogeneous as well as non-homogeneous nonlinear Lane-Emden type equations. Using proposed computational method given nonlinear equation is converted into a set of nonlinear algebraic equations whose solution gives the approximate solution to the Lane-Emden type equation. Various nonlinear cases of Lane-Emden type equations like standard Lane-Emden equation, the isothermal gas spheres equation and white-dwarf equation are discussed. Results are compared with some well-known numerical methods and it is observed that our results are more accurate.

Numerical study of fractional nonlinear Schrödinger equations.

PubMed

Klein, Christian; Sparber, Christof; Markowich, Peter

2014-12-08

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.

Numerical study of fractional nonlinear Schrödinger equations

PubMed Central

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 hierarchy of nonlinearly dispersive generalized Korteweg - de Vries evolution equations

DOE PAGES

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.

Nonlinear Solver Approaches for the Diffusive Wave Approximation to the Shallow Water Equations

NASA Astrophysics Data System (ADS)

Collier, N.; Knepley, M.

2015-12-01

The diffusive wave approximation to the shallow water equations (DSW) is a doubly-degenerate, nonlinear, parabolic partial differential equation used to model overland flows. Despite its challenges, the DSW equation has been extensively used to model the overland flow component of various integrated surface/subsurface models. The equation's complications become increasingly problematic when ponding occurs, a feature which becomes pervasive when solving on large domains with realistic terrain. In this talk I discuss the various forms and regularizations of the DSW equation and highlight their effect on the solvability of the nonlinear system. In addition to this analysis, I present results of a numerical study which tests the applicability of a class of composable nonlinear algebraic solvers recently added to the Portable, Extensible, Toolkit for Scientific Computation (PETSc).

Dispersive optical soliton solutions for the hyperbolic and cubic-quintic nonlinear Schrödinger equations via the extended sinh-Gordon equation expansion method

NASA Astrophysics Data System (ADS)

Seadawy, Aly R.; Kumar, Dipankar; Chakrabarty, Anuz Kumar

2018-05-01

The (2+1)-dimensional hyperbolic and cubic-quintic nonlinear Schrödinger equations describe the propagation of ultra-short pulses in optical fibers of nonlinear media. By using an extended sinh-Gordon equation expansion method, some new complex hyperbolic and trigonometric functions prototype solutions for two nonlinear Schrödinger equations were derived. The acquired new complex hyperbolic and trigonometric solutions are expressed by dark, bright, combined dark-bright, singular and combined singular solitons. The obtained results are more compatible than those of other applied methods. The extended sinh-Gordon equation expansion method is a more powerful and robust mathematical tool for generating new optical solitary wave solutions for many other nonlinear evolution equations arising in the propagation of optical pulses.

Nonlinear anomalous diffusion equation and fractal dimension: exact generalized Gaussian solution.

PubMed

Pedron, I T; Mendes, R S; Malacarne, L C; Lenzi, E K

2002-04-01

In this work we incorporate, in a unified way, two anomalous behaviors, the power law and stretched exponential ones, by considering the radial dependence of the N-dimensional nonlinear diffusion equation partial differential rho/ partial differential t=nabla.(Knablarho(nu))-nabla.(muFrho)-alpharho, where K=Dr(-theta), nu, theta, mu, and D are real parameters, F is the external force, and alpha is a time-dependent source. This equation unifies the O'Shaughnessy-Procaccia anomalous diffusion equation on fractals (nu=1) and the spherical anomalous diffusion for porous media (theta=0). An exact spherical symmetric solution of this nonlinear Fokker-Planck equation is obtained, leading to a large class of anomalous behaviors. Stationary solutions for this Fokker-Planck-like equation are also discussed by introducing an effective potential.

A quadrature based method of moments for nonlinear Fokker-Planck equations

NASA Astrophysics Data System (ADS)

Otten, Dustin L.; Vedula, Prakash

2011-09-01

Fokker-Planck equations which are nonlinear with respect to their probability densities and occur in many nonequilibrium systems relevant to mean field interaction models, plasmas, fermions and bosons can be challenging to solve numerically. To address some underlying challenges, we propose the application of the direct quadrature based method of moments (DQMOM) for efficient and accurate determination of transient (and stationary) solutions of nonlinear Fokker-Planck equations (NLFPEs). In DQMOM, probability density (or other distribution) functions are represented using a finite collection of Dirac delta functions, characterized by quadrature weights and locations (or abscissas) that are determined based on constraints due to evolution of generalized moments. Three particular examples of nonlinear Fokker-Planck equations considered in this paper include descriptions of: (i) the Shimizu-Yamada model, (ii) the Desai-Zwanzig model (both of which have been developed as models of muscular contraction) and (iii) fermions and bosons. Results based on DQMOM, for the transient and stationary solutions of the nonlinear Fokker-Planck equations, have been found to be in good agreement with other available analytical and numerical approaches. It is also shown that approximate reconstruction of the underlying probability density function from moments obtained from DQMOM can be satisfactorily achieved using a maximum entropy method.

A nonlinear wave equation in nonadiabatic flame propagation

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

Booty, M.R.; Matalon, M.; Matkowsky, B.J.

1988-06-01

The authors derive a nonlinear wave equation from the diffusional thermal model of gaseous combustion to describe the evolution of a flame front. The equation arises as a long wave theory, for values of the volumeric heat loss in a neighborhood of the extinction point (beyond which planar uniformly propagating flames cease to exist), and for Lewis numbers near the critical value beyond which uniformly propagating planar flames lose stability via a degenerate Hopf bifurcation. Analysis of the equation suggests the possibility of a singularity developing in finite time.

Lichnerowicz-type equations with sign-changing nonlinearities on complete manifolds with boundary

NASA Astrophysics Data System (ADS)

Albanese, Guglielmo; Rigoli, Marco

2017-12-01

We prove an existence theorem for positive solutions to Lichnerowicz-type equations on complete manifolds with boundary (M , ∂ M , 〈 , 〉) and nonlinear Neumann conditions. This kind of nonlinear problems arise quite naturally in the study of solutions for the Einstein-scalar field equations of General Relativity in the framework of the so called Conformal Method.

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

NASA Astrophysics Data System (ADS)

Przedborski, Michelle; Anco, Stephen C.

2017-09-01

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

Some Remarks on Similarity and Soliton Solutions of Nonlinear Klein-Gordon Equation

NASA Astrophysics Data System (ADS)

Tajiri, Masayoshi

1984-11-01

The three-dimensional nonlinear Klein-Gordon [, Higgs field and Yang-Milles] (3D-KG [, H and YM]) equation is first reduced to the 2D nonlinear Schrödinger (2D-NLS) and 2D-KG [, H and YM] equations, and secondly to the 1D-NLS and 1D-KG [, H and YM] equations by similarity transformations. It is shown that similar type soliton solutions of the 3D-KG, H and YM equations, which have singularity on a plane in (x, y, z, t) space, are obtained by substituting the soliton solutions of the 1D-NLS or 1D-KG (or H) equation into the similarity transformations. The soliton solutions of the YM equation are also investigated.

Initial-value problem for the Gardner equation applied to nonlinear internal waves

NASA Astrophysics Data System (ADS)

Rouvinskaya, Ekaterina; Kurkina, Oxana; Kurkin, Andrey; Talipova, Tatiana; Pelinovsky, Efim

2017-04-01

The Gardner equation is a fundamental mathematical model for the description of weakly nonlinear weakly dispersive internal waves, when cubic nonlinearity cannot be neglected. Within this model coefficients of quadratic and cubic nonlinearity can both be positive as well as negative, depending on background conditions of the medium, where waves propagate (sea water density stratification, shear flow profile) [Rouvinskaya et al., 2014, Kurkina et al., 2011, 2015]. For the investigation of weakly dispersive behavior in the framework of nondimensional Gardner equation with fixed (positive) sign of quadratic nonlinearity and positive or negative cubic nonlinearity {eq1} partial η/partial t+6η( {1± η} )partial η/partial x+partial ^3η/partial x^3=0, } the series of numerical experiments of initial-value problem was carried out for evolution of a bell-shaped impulse of negative polarity (opposite to the sign of quadratic nonlinear coefficient): {eq2} η(x,t=0)=-asech2 ( {x/x0 } ), for which amplitude a and width x0 was varied. Similar initial-value problem was considered in the paper [Trillo et al., 2016] for the Korteweg - de Vries equation. For the Gardner equation with different signs of cubic nonlinearity the initial-value problem for piece-wise constant initial condition was considered in detail in [Grimshaw et al., 2002, 2010]. It is widely known, for example, [Pelinovsky et al., 2007], that the Gardner equation (1) with negative cubic nonlinearity has a family of classic solitary wave solutions with only positive polarity,and with limiting amplitude equal to 1. Therefore evolution of impulses (2) of negative polarity (whose amplitudes a were varied from 0.1 to 3, and widths at the level of a/2 were equal to triple width of solitons with the same amplitude for a 1) was going on a universal scenario with the generation of nonlinear Airy wave. For the Gardner equation (1) with the positive cubic nonlinearity coefficient there exist two one-parametric families of

Asymptotic integration algorithms for nonhomogeneous, nonlinear, first order, ordinary differential equations

NASA Technical Reports Server (NTRS)

Walker, K. P.; Freed, A. D.

1991-01-01

New methods for integrating systems of stiff, nonlinear, first order, ordinary differential equations are developed by casting the differential equations into integral form. Nonlinear recursive relations are obtained that allow the solution to a system of equations at time t plus delta t to be obtained in terms of the solution at time t in explicit and implicit forms. Examples of accuracy obtained with the new technique are given by considering systems of nonlinear, first order equations which arise in the study of unified models of viscoplastic behaviors, the spread of the AIDS virus, and predator-prey populations. In general, the new implicit algorithm is unconditionally stable, and has a Jacobian of smaller dimension than that which is acquired by current implicit methods, such as the Euler backward difference algorithm; yet, it gives superior accuracy. The asymptotic explicit and implicit algorithms are suitable for solutions that are of the growing and decaying exponential kinds, respectively, whilst the implicit Euler-Maclaurin algorithm is superior when the solution oscillates, i.e., when there are regions in which both growing and decaying exponential solutions exist.

A Bifurcation Problem for a Nonlinear Partial Differential Equation of Parabolic Type,

DTIC Science & Technology

NONLINEAR DIFFERENTIAL EQUATIONS, INTEGRATION), (*PARTIAL DIFFERENTIAL EQUATIONS, BOUNDARY VALUE PROBLEMS), BANACH SPACE , MAPPING (TRANSFORMATIONS), SET THEORY, TOPOLOGY, ITERATIONS, STABILITY, THEOREMS

Dark Solitons for the Defocusing Cubic Nonlinear Schrödinger Equation with the Spatially Periodic Potential and Nonlinearity

NASA Astrophysics Data System (ADS)

Yan, Zhen-Ya; Yan, Fang-Chi

2015-09-01

We study the existence of dark solitons of the defocusing cubic nonlinear Schrödinger (NLS) eqaution with the spatially-periodic potential and nonlinearity. Firstly, we propose six families of upper and lower solutions of the dynamical systems arising from the stationary defocusing NLS equation. Secondly, by regarding a dark soliton as a heteroclinic orbit of the Poincaré map, we present some constraint conditions for the periodic potential and nonlinearity to show the existence of stationary dark solitons of the defocusing NLS equation for six different cases in terms of the theory of strict lower and upper solutions and the dynamics of planar homeomorphisms. Finally, we give the explicit dark solitons of the defocusing NLS equation with the chosen periodic potential and nonlinearity. Supported by the National Natural Science Foundation of China under Grant No. 61178091, the National Key Basic Research Program of China under Grant No. 2011CB302400, and the Open Project Program of State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, China under Grant No. Y4KF211CJ1

New extended (G'/G)-expansion method to solve nonlinear evolution equation: the (3 + 1)-dimensional potential-YTSF equation.

PubMed

Roshid, Harun-Or-; Akbar, M Ali; Alam, Md Nur; Hoque, Md Fazlul; Rahman, Nizhum

2014-01-01

In this article, a new extended (G'/G) -expansion method has been proposed for constructing more general exact traveling wave solutions of nonlinear evolution equations with the aid of symbolic computation. In order to illustrate the validity and effectiveness of the method, we pick the (3 + 1)-dimensional potential-YTSF equation. As a result, abundant new and more general exact solutions have been achieved of this equation. It has been shown that the proposed method provides a powerful mathematical tool for solving nonlinear wave equations in applied mathematics, engineering and mathematical physics.

Nonlinear Field Equations and Solitons as Particles

NASA Astrophysics Data System (ADS)

Maccari, Attilio

2006-05-01

Profound advances have recently interested nonlinear field theories and their exact or approximate solutions. We review the last results and point out some important unresolved questions. It is well known that quantum field theories are based upon Fourier series and the identification of plane waves with free particles. On the contrary, nonlinear field theories admit the existence of coherent solutions (dromions, solitons and so on). Moreover, one can construct lower dimensional chaotic patterns, periodic-chaotic patterns, chaotic soliton and dromion patterns. In a similar way, fractal dromion and lump patterns as well as stochastic fractal excitations can appear in the solution. We discuss in some detail a nonlinear Dirac field and a spontaneous symmetry breaking model that are reduced by means of the asymptotic perturbation method to a system of nonlinear evolution equations integrable via an appropriate change of variables. Their coherent, chaotic and fractal solutions are examined in some detail. Finally, we consider the possible identification of some types of coherent solutions with extended particles along the de Broglie-Bohm theory. However, the last findings suggest an inadequacy of the particle concept that appears only as a particular case of nonlinear field theories excitations.

Concatenons as the solutions for non-linear partial differential equations

NASA Astrophysics Data System (ADS)

Kudryashov, N. A.; Volkov, A. K.

2017-07-01

New class of solutions for nonlinear partial differential equations is introduced. We call them the concaten solutions. As an example we consider equations for the description of wave processes in the Fermi-Pasta-Ulam mass chain and construct the concatenon solutions for these equation. Stability of the concatenon-type solutions is investigated numerically. Interaction between the concatenon and solitons is discussed.

Nonlinear tunneling of optical soliton in 3 coupled NLS equation with symbolic computation

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

Mani Rajan, M.S., E-mail: senthilmanirajanofc@gmail.com; Mahalingam, A.; Uthayakumar, A.

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,more » 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.« less

Nonlocal nonlinear Schrödinger equations and their soliton solutions

NASA Astrophysics Data System (ADS)

Gürses, Metin; Pekcan, Aslı

2018-05-01

We study standard and nonlocal nonlinear Schrödinger (NLS) equations obtained from the coupled NLS system of equations (Ablowitz-Kaup-Newell-Segur (AKNS) equations) by using standard and nonlocal reductions, respectively. By using the Hirota bilinear method, we first find soliton solutions of the coupled NLS system of equations; then using the reduction formulas, we find the soliton solutions of the standard and nonlocal NLS equations. We give examples for particular values of the parameters and plot the function |q(t, x)|2 for the standard and nonlocal NLS equations.

A nonlinear viscoelastic constitutive equation - Yield predictions in multiaxial deformations

NASA Technical Reports Server (NTRS)

Shay, R. M., Jr.; Caruthers, J. M.

1987-01-01

Yield stress predictions of a nonlinear viscoelastic constitutive equation for amorphous polymer solids have been obtained and are compared with the phenomenological von Mises yield criterion. Linear viscoelasticity theory has been extended to include finite strains and a material timescale that depends on the instantaneous temperature, volume, and pressure. Results are presented for yield and the correct temperature and strain-rate dependence in a variety of multiaxial deformations. The present nonlinear viscoelastic constitutive equation can be formulated in terms of either a Cauchy or second Piola-Kirchhoff stress tensor, and in terms of either atmospheric or hydrostatic pressure.

On the structure of nonlinear constitutive equations for fiber reinforced composites

NASA Technical Reports Server (NTRS)

Jansson, Stefan

1992-01-01

The structure of constitutive equations for nonlinear multiaxial behavior of transversely isotropic fiber reinforced metal matrix composites subject to proportional loading was investigated. Results from an experimental program were combined with numerical simulations of the composite behavior for complex stress to reveal the full structure of the equations. It was found that the nonlinear response can be described by a quadratic flow-potential, based on the polynomial stress invariants, together with a hardening rule that is dominated by two different hardening mechanisms.

Solutions to Some Nonlinear Equations from Nonmetric Data.

ERIC Educational Resources Information Center

Rule, Stanley J.

1979-01-01

A method to provide estimates of parameters of specified nonlinear equations from ordinal data generated from a crossed design is presented. The statistical basis for the method, called NOPE (nonmetric parameter estimation), as well as examples using artifical data, are presented. (Author/JKS)

Nonlinear static and dynamic analysis of beam structures using fully intrinsic equations

NASA Astrophysics Data System (ADS)

Sotoudeh, Zahra

2011-07-01

Beams are structural members with one dimension much larger than the other two. Examples of beams include propeller blades, helicopter rotor blades, and high aspect-ratio aircraft wings in aerospace engineering; shafts and wind turbine blades in mechanical engineering; towers, highways and bridges in civil engineering; and DNA modeling in biomedical engineering. Beam analysis includes two sets of equations: a generally linear two-dimensional problem over the cross-sectional plane and a nonlinear, global one-dimensional analysis. This research work deals with a relatively new set of equations for one-dimensional beam analysis, namely the so-called fully intrinsic equations. Fully intrinsic equations comprise a set of geometrically exact, nonlinear, first-order partial differential equations that is suitable for analyzing initially curved and twisted anisotropic beams. A fully intrinsic formulation is devoid of displacement and rotation variables, making it especially attractive because of the absence of singularities, infinite-degree nonlinearities, and other undesirable features associated with finite rotation variables. In spite of the advantages of these equations, using them with certain boundary conditions presents significant challenges. This research work will take a broad look at these challenges of modeling various boundary conditions when using the fully intrinsic equations. Hopefully it will clear the path for wider and easier use of the fully intrinsic equations in future research. This work also includes application of fully intrinsic equations in structural analysis of joined-wing aircraft, different rotor blade configuration and LCO analysis of HALE aircraft.

Analytical approximate solutions for a general class of nonlinear delay differential equations.

PubMed

Căruntu, Bogdan; Bota, Constantin

2014-01-01

We use the polynomial least squares method (PLSM), which allows us to compute analytical approximate polynomial solutions for a very general class of strongly nonlinear delay differential equations. The method is tested by computing approximate solutions for several applications including the pantograph equations and a nonlinear time-delay model from biology. The accuracy of the method is illustrated by a comparison with approximate solutions previously computed using other methods.

Nonlinear Schrödinger equation in foundations: summary of 4 catches

NASA Astrophysics Data System (ADS)

Diósi, Lajos

2016-03-01

Fundamental modifications of the standard Schrödinger equation by additional nonlinear terms have been considered for various purposes over the recent decades. It came as a surprise when, inverting Abner Shimonyi's observation of “peaceful coexistence” between standard quantum mechanics and relativity, N. Gisin proved in 1990 that any (deterministic) nonlinear Schrödinger equation would allow for superluminal communication. This is by now the most spectacular and best known anomaly. We discuss further anomalies, simple but foundational, less spectacular but not less dramatic.

Oscillation theorems for second order nonlinear forced differential equations.

PubMed

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.

Nonlinear Schrödinger equations with single power nonlinearity and harmonic potential

NASA Astrophysics Data System (ADS)

Cipolatti, R.; de Macedo Lira, Y.; Trallero-Giner, C.

2018-03-01

We consider a generalized nonlinear Schrödinger equation (GNLS) with a single power nonlinearity of the form λ ≤ft\\vert \\varphi \\right\\vert p , with p  >  0 and λ\\in{R} , in the presence of a harmonic confinement. We report the conditions that p and λ must fulfill for the existence and uniqueness of ground states of the GNLS. We discuss the Cauchy problem and summarize which conditions are required for the nonlinear term λ ≤ft\\vert \\varphi \\right\\vert p to render the ground state solutions orbitally stable. Based on a new variational method we provide exact formulæ for the minimum energy for each index p and the changing range of values of the nonlinear parameter λ. Also, we report an approximate close analytical expression for the ground state energy, performing a comparative analysis of the present variational calculations with those obtained by a generalized Thomas-Fermi approach, and soliton solutions for the respective ranges of p and λ where these solutions can be implemented to describe the minimum energy.

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

NASA Astrophysics Data System (ADS)

Seadawy, Aly R.

2017-09-01

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

Lump solutions to nonlinear partial differential equations via Hirota bilinear forms

NASA Astrophysics Data System (ADS)

Ma, Wen-Xiu; Zhou, Yuan

2018-02-01

Lump solutions are analytical rational function solutions localized in all directions in space. We analyze a class of lump solutions, generated from quadratic functions, to nonlinear partial differential equations. The basis of success is the Hirota bilinear formulation and the primary object is the class of positive multivariate quadratic functions. A complete determination of quadratic functions positive in space and time is given, and positive quadratic functions are characterized as sums of squares of linear functions. Necessary and sufficient conditions for positive quadratic functions to solve Hirota bilinear equations are presented, and such polynomial solutions yield lump solutions to nonlinear partial differential equations under the dependent variable transformations u = 2(ln ⁡ f) x and u = 2(ln ⁡ f) xx, where x is one spatial variable. Applications are made for a few generalized KP and BKP equations.

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.

Symmetry reduction and exact solutions of two higher-dimensional nonlinear evolution equations.

PubMed

Gu, Yongyi; Qi, Jianming

2017-01-01

In this paper, symmetries and symmetry reduction of two higher-dimensional nonlinear evolution equations (NLEEs) are obtained by Lie group method. These NLEEs play an important role in nonlinear sciences. We derive exact solutions to these NLEEs via the [Formula: see text]-expansion method and complex method. Five types of explicit function solutions are constructed, which are rational, exponential, trigonometric, hyperbolic and elliptic function solutions of the variables in the considered equations.

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)

On the numerical treatment of nonlinear source terms in reaction-convection equations

NASA Technical Reports Server (NTRS)

Lafon, A.; Yee, H. C.

1992-01-01

The objectives of this paper are to investigate how various numerical treatments of the nonlinear source term in a model reaction-convection equation can affect the stability of steady-state numerical solutions and to show under what conditions the conventional linearized analysis breaks down. The underlying goal is to provide part of the basic building blocks toward the ultimate goal of constructing suitable numerical schemes for hypersonic reacting flows, combustions and certain turbulence models in compressible Navier-Stokes computations. It can be shown that nonlinear analysis uncovers much of the nonlinear phenomena which linearized analysis is not capable of predicting in a model reaction-convection equation.

Traveling wave solutions of the nonlinear Schrödinger equation

NASA Astrophysics Data System (ADS)

Akbari-Moghanjoughi, M.

2017-10-01

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

Soliton solutions of an integrable nonlinear Schrödinger equation with quintic terms.

PubMed

Chowdury, A; Kedziora, D J; Ankiewicz, A; Akhmediev, N

2014-09-01

We present the fifth-order equation of the nonlinear Schrödinger hierarchy. This integrable partial differential equation contains fifth-order dispersion and nonlinear terms related to it. We present the Lax pair and use Darboux transformations to derive exact expressions for the most representative soliton solutions. This set includes two-soliton collisions and the degenerate case of the two-soliton solution, as well as beating structures composed of two or three solitons. Ultimately, the new quintic operator and the terms it adds to the standard nonlinear Schrödinger equation (NLSE) are found to primarily affect the velocity of solutions, with complicated flow-on effects. Furthermore, we present a new structure, composed of coincident equal-amplitude solitons, which cannot exist for the standard NLSE.

Coupled equations of electromagnetic waves in nonlinear metamaterial waveguides.

PubMed

Azari, Mina; Hatami, Mohsen; Meygoli, Vahid; Yousefi, Elham

2016-11-01

Over the past decades, scientists have presented ways to manipulate the macroscopic properties of a material at levels unachieved before, and called them metamaterials. This research can be considered an important step forward in electromagnetics and optics. In this study, higher-order nonlinear coupled equations in a special kind of metamaterial waveguides (a planar waveguide with metamaterial core) will be derived from both electric and magnetic components of the transverse electric mode of electromagnetic pulse propagation. On the other hand, achieving the refractive index in this research is worthwhile. It is also shown that the coupled equations are not symmetric with respect to the electric and magnetic fields, unlike these kinds of equations in fiber optics and dielectric waveguides. Simulations on the propagation of a fundamental soliton pulse in a nonlinear metamaterial waveguide near the resonance frequency (a little lower than the magnetic resonant frequency) are performed to study its behavior. These pulses are recommended to practice in optical communications in controlled switching by external voltage, even in low power.

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

NASA Astrophysics Data System (ADS)

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

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

Schroedinger's immortal cat

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

Peres, A.

1988-01-01

The purpose of this paper is to review and clarify the quantum measurement problem. The latter originates in the ambivalent nature of the observer: Although the observer is not described by the Schroedinger equation, it should nevertheless be possible to quantize him and include him in the wave function if quantum theory is universally valid. The problem is to prove that no contradiction may arise in these two conflicting descriptions. The proof invokes the notion of irreversibility. The validity of the latter is questionable, because the standard rationale for classical irreversibility, namely mixing and coarse graining, does not apply tomore » quantum theory. There is no chaos in a closed, finite quantum system. However, when a system is large enough, it cannot be perfectly isolated from it environment, namely from external (or even internal) degrees of freedom which are not fully accounted for in the Hamiltonian of that system. As a consequence, the long-range evolution of such a quantum system is essentially unpredictable. It follows that the notion of irreversibility is a valid one in quantum theory and the measurement problem can be brought to a satisfactory solution.« less

Bright and singular soliton solutions of the conformable time-fractional Klein-Gordon equations with different nonlinearities

NASA Astrophysics Data System (ADS)

Hosseini, Kamyar; Mayeli, Peyman; Ansari, Reza

2018-07-01

Finding the exact solutions of nonlinear fractional differential equations has gained considerable attention, during the past two decades. In this paper, the conformable time-fractional Klein-Gordon equations with quadratic and cubic nonlinearities are studied. Several exact soliton solutions, including the bright (non-topological) and singular soliton solutions are formally extracted by making use of the ansatz method. Results demonstrate that the method can efficiently handle the time-fractional Klein-Gordon equations with different nonlinearities.

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.

Entropic nonadditivity, H theorem, and nonlinear Klein-Kramers equations.

PubMed

Dos Santos, M A F; Lenzi, E K

2017-11-01

We use the H theorem to establish the entropy and the entropic additivity law for a system composed of subsystems, with the dynamics governed by the Klein-Kramers equations, by considering relations among the dynamics of these subsystems and their entropies. We start considering the subsystems governed by linear Klein-Kramers equations and verify that the Boltzmann-Gibbs entropy is appropriated to this dynamics, leading us to the standard entropic additivity, S_{BG}^{(1∪2)}=S_{BG}^{1}+S_{BG}^{2}, consistent with the fact that the distributions of the subsystem are independent. We then extend the dynamics of these subsystems to independent nonlinear Klein-Kramers equations. For this case, the results show that the H theorem is verified for a generalized entropy, which does not preserve the standard entropic additivity for independent distributions. In this scenario, consistent results are obtained when a suitable coupling among the nonlinear Klein-Kramers equations is considered, in which each subsystem modifies the other until an equilibrium state is reached. This dynamics, for the subsystems, results in the Tsallis entropy for the system and, consequently, verifies the relation S_{q}^{(1∪2)}=S_{q}^{1}+S_{q}^{2}+(1-q)S_{q}^{1}S_{q}^{2}/k, which is a nonadditive entropic relation.

Regularity for Fully Nonlinear Elliptic Equations with Oblique Boundary Conditions

NASA Astrophysics Data System (ADS)

Li, Dongsheng; Zhang, Kai

2018-06-01

In this paper, we obtain a series of regularity results for viscosity solutions of fully nonlinear elliptic equations with oblique derivative boundary conditions. In particular, we derive the pointwise C α, C 1,α and C 2,α regularity. As byproducts, we also prove the A-B-P maximum principle, Harnack inequality, uniqueness and solvability of the equations.

Higher symmetries and exact solutions of linear and nonlinear Schr{umlt o}dinger equation

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

Fushchych, W.I.; Nikitin, A.G.

1997-11-01

A new approach for the analysis of partial differential equations is developed which is characterized by a simultaneous use of higher and conditional symmetries. Higher symmetries of the Schr{umlt o}dinger equation with an arbitrary potential are investigated. Nonlinear determining equations for potentials are solved using reductions to Weierstrass, Painlev{acute e}, and Riccati forms. Algebraic properties of higher order symmetry operators are analyzed. Combinations of higher and conditional symmetries are used to generate families of exact solutions of linear and nonlinear Schr{umlt o}dinger equations. {copyright} {ital 1997 American Institute of Physics.}

Periodic solutions for one dimensional wave equation with bounded nonlinearity

NASA Astrophysics Data System (ADS)

Ji, Shuguan

2018-05-01

This paper is concerned with the periodic solutions for the one dimensional nonlinear wave equation with either constant or variable coefficients. The constant coefficient model corresponds to the classical wave equation, while the variable coefficient model arises from the forced vibrations of a nonhomogeneous string and the propagation of seismic waves in nonisotropic media. For finding the periodic solutions of variable coefficient wave equation, it is usually required that the coefficient u (x) satisfies ess infηu (x) > 0 with ηu (x) = 1/2 u″/u - 1/4 (u‧/u)2, which actually excludes the classical constant coefficient model. For the case ηu (x) = 0, it is indicated to remain an open problem by Barbu and Pavel (1997) [6]. In this work, for the periods having the form T = 2p-1/q (p , q are positive integers) and some types of boundary value conditions, we find some fundamental properties for the wave operator with either constant or variable coefficients. Based on these properties, we obtain the existence of periodic solutions when the nonlinearity is monotone and bounded. Such nonlinearity may cross multiple eigenvalues of the corresponding wave operator. In particular, we do not require the condition ess infηu (x) > 0.

Model Equation for Acoustic Nonlinear Measurement of Dispersive Specimens at High Frequency

NASA Astrophysics Data System (ADS)

Zhang, Dong; Kushibiki, Junichi; Zou, Wei

2006-10-01

We present a theoretical model for acoustic nonlinearity measurement of dispersive specimens at high frequency. The nonlinear Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation governs the nonlinear propagation in the SiO2/specimen/SiO2 multi-layer medium. The dispersion effect is considered in a special manner by introducing the frequency-dependant sound velocity in the KZK equation. Simple analytic solutions are derived by applying the superposition technique of Gaussian beams. The solutions are used to correct the diffraction and dispersion effects in the measurement of acoustic nonlinearity of cottonseed oil in the frequency range of 33-96 MHz. Regarding two different ultrasonic devices, the accuracies of the measurements are improved to ±2.0% and ±1.3% in comparison with ±9.8% and ±2.9% obtained from the previous plane wave model.

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…

Analytical-numerical solution of a nonlinear integrodifferential equation in econometrics

NASA Astrophysics Data System (ADS)

Kakhktsyan, V. M.; Khachatryan, A. Kh.

2013-07-01

A mixed problem for a nonlinear integrodifferential equation arising in econometrics is considered. An analytical-numerical method is proposed for solving the problem. Some numerical results are presented.

Polynomial elimination theory and non-linear stability analysis for the Euler equations

NASA Technical Reports Server (NTRS)

Kennon, S. R.; Dulikravich, G. S.; Jespersen, D. C.

1986-01-01

Numerical methods are presented that exploit the polynomial properties of discretizations of the Euler equations. It is noted that most finite difference or finite volume discretizations of the steady-state Euler equations produce a polynomial system of equations to be solved. These equations are solved using classical polynomial elimination theory, with some innovative modifications. This paper also presents some preliminary results of a new non-linear stability analysis technique. This technique is applicable to determining the stability of polynomial iterative schemes. Results are presented for applying the elimination technique to a one-dimensional test case. For this test case, the exact solution is computed in three iterations. The non-linear stability analysis is applied to determine the optimal time step for solving Burgers' equation using the MacCormack scheme. The estimated optimal time step is very close to the time step that arises from a linear stability analysis.

Traveling wave and exact solutions for the perturbed nonlinear Schrödinger equation with Kerr law nonlinearity

NASA Astrophysics Data System (ADS)

Akram, Ghazala; Mahak, Nadia

2018-06-01

The nonlinear Schrödinger equation (NLSE) with the aid of three order dispersion terms is investigated to find the exact solutions via the extended (G'/G2)-expansion method and the first integral method. Many exact traveling wave solutions, such as trigonometric, hyperbolic, rational, soliton and complex function solutions, are characterized with some free parameters of the problem studied. It is corroborated that the proposed techniques are manageable, straightforward and powerful tools to find the exact solutions of nonlinear partial differential equations (PDEs). Some figures are plotted to describe the propagation of traveling wave solutions expressed by the hyperbolic functions, trigonometric functions and rational functions.

Optical solitons, nonlinear self-adjointness and conservation laws for the cubic nonlinear Shrödinger's equation with repulsive delta potential

NASA Astrophysics Data System (ADS)

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

2017-11-01

In this paper, the complex envelope function ansatz method is used to acquire the optical solitons to the cubic nonlinear Shrödinger's equation with repulsive delta potential (δ-NLSE). The method reveals dark and bright optical solitons. The necessary constraint conditions which guarantee the existence of the solitons are also presented. We studied the δ-NLSE by analyzing a system of partial differential equations (PDEs) obtained by decomposing the equation into real and imaginary components. We derive the Lie point symmetry generators of the system and prove that the system is nonlinearly self-adjoint with an explicit form of a differential substitution satisfying the nonlinear self-adjoint condition. Then we use these facts to establish a set of conserved vectors for the system using the general Cls theorem presented by Ibragimov. Some interesting figures for the acquired solutions are also presented.

Spectral Target Detection using Schroedinger Eigenmaps

NASA Astrophysics Data System (ADS)

Dorado-Munoz, Leidy P.

Applications of optical remote sensing processes include environmental monitoring, military monitoring, meteorology, mapping, surveillance, etc. Many of these tasks include the detection of specific objects or materials, usually few or small, which are surrounded by other materials that clutter the scene and hide the relevant information. This target detection process has been boosted lately by the use of hyperspectral imagery (HSI) since its high spectral dimension provides more detailed spectral information that is desirable in data exploitation. Typical spectral target detectors rely on statistical or geometric models to characterize the spectral variability of the data. However, in many cases these parametric models do not fit well HSI data that impacts the detection performance. On the other hand, non-linear transformation methods, mainly based on manifold learning algorithms, have shown a potential use in HSI transformation, dimensionality reduction and classification. In target detection, non-linear transformation algorithms are used as preprocessing techniques that transform the data to a more suitable lower dimensional space, where the statistical or geometric detectors are applied. One of these non-linear manifold methods is the Schroedinger Eigenmaps (SE) algorithm that has been introduced as a technique for semi-supervised classification. The core tool of the SE algorithm is the Schroedinger operator that includes a potential term that encodes prior information about the materials present in a scene, and enables the embedding to be steered in some convenient directions in order to cluster similar pixels together. A completely novel target detection methodology based on SE algorithm is proposed for the first time in this thesis. The proposed methodology does not just include the transformation of the data to a lower dimensional space but also includes the definition of a detector that capitalizes on the theory behind SE. The fact that target pixels and

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.

Path-Following Solutions Of Nonlinear Equations

NASA Technical Reports Server (NTRS)

Barger, Raymond L.; Walters, Robert W.

1989-01-01

Report describes some path-following techniques for solution of nonlinear equations and compares with other methods. Use of multipurpose techniques applicable at more than one stage of path-following computation results in system relatively simple to understand, program, and use. Comparison of techniques with method of parametric differentiation (MPD) reveals definite advantages for path-following methods. Emphasis in investigation on multiuse techniques being applied at more than one stage of path-following computation. Incorporation of multipurpose techniques results in concise computer code relatively simple to use.

Solvability of a Nonlinear Integral Equation in Dynamical String Theory

NASA Astrophysics Data System (ADS)

Khachatryan, A. Kh.; Khachatryan, Kh. A.

2018-04-01

We investigate an integral equation of the convolution type with a cubic nonlinearity on the entire real line. This equation has a direct application in open-string field theory and in p-adic string theory and describes nonlocal interactions. We prove that there exists a one-parameter family of bounded monotonic solutions and calculate the limits of solutions constructed at infinity.

Nonlinear Feedback Controllers and Compensators: A State-Dependent Riccati Equation Approach

DTIC Science & Technology

2003-01-01

Nonlinear Feedback Controllers and Compensators: A State-Dependent Riccati Equation Approach H. T. Banks∗ B. M. Lewis † H. T. Tran‡ Department of...Mathematics Center for Research in Scientific Computation North Carolina State University Raleigh, NC 27695 Abstract State-dependent Riccati equation ...estimating the solution of the Hamilton- Jacobi-Bellman (HJB) equation can be found in a comprehensive review article [5]. Each of these ∗htbanks

Use of Picard and Newton iteration for solving nonlinear ground water flow equations

USGS Publications Warehouse

Mehl, S.

2006-01-01

This study examines the use of Picard and Newton iteration to solve the nonlinear, saturated ground water flow equation. Here, a simple three-node problem is used to demonstrate the convergence difficulties that can arise when solving the nonlinear, saturated ground water flow equation in both homogeneous and heterogeneous systems with and without nonlinear boundary conditions. For these cases, the characteristic types of convergence patterns are examined. Viewing these convergence patterns as orbits of an attractor in a dynamical system provides further insight. It is shown that the nonlinearity that arises from nonlinear head-dependent boundary conditions can cause more convergence difficulties than the nonlinearity that arises from flow in an unconfined aquifer. Furthermore, the effects of damping on both convergence and convergence rate are investigated. It is shown that no single strategy is effective for all problems and how understanding pitfalls and merits of several methods can be helpful in overcoming convergence difficulties. Results show that Picard iterations can be a simple and effective method for the solution of nonlinear, saturated ground water flow problems.

Nonlinear Schrödinger equations for Bose-Einstein condensates

NASA Astrophysics Data System (ADS)

Galati, Luigi; Zheng, Shijun

2013-10-01

The Gross-Pitaevskii equation, or more generally the nonlinear Schrödinger equation, models the Bose-Einstein condensates in a macroscopic gaseous superfluid wave-matter state in ultra-cold temperature. We provide analytical study of the NLS with L2 initial data in order to understand propagation of the defocusing and focusing waves for the BEC mechanism in the presence of electromagnetic fields. Numerical simulations are performed for the two-dimensional GPE with anisotropic quadratic potentials.

New exact solutions of the Tzitzéica-type equations in non-linear optics using the expa function method

NASA Astrophysics Data System (ADS)

Hosseini, K.; Ayati, Z.; Ansari, R.

2018-04-01

One specific class of non-linear evolution equations, known as the Tzitzéica-type equations, has received great attention from a group of researchers involved in non-linear science. In this article, new exact solutions of the Tzitzéica-type equations arising in non-linear optics, including the Tzitzéica, Dodd-Bullough-Mikhailov and Tzitzéica-Dodd-Bullough equations, are obtained using the expa function method. The integration technique actually suggests a useful and reliable method to extract new exact solutions of a wide range of non-linear evolution equations.

Davidenko’s Method for the Solution of Nonlinear Operator Equations.

DTIC Science & Technology

NONLINEAR DIFFERENTIAL EQUATIONS, NUMERICAL INTEGRATION), OPERATORS(MATHEMATICS), BANACH SPACE , MAPPING (TRANSFORMATIONS), NUMERICAL METHODS AND PROCEDURES, INTEGRALS, SET THEORY, CONVERGENCE, MATRICES(MATHEMATICS)

Modified Taylor series method for solving nonlinear differential equations with mixed boundary conditions defined on finite intervals.

PubMed

Vazquez-Leal, Hector; Benhammouda, Brahim; Filobello-Nino, Uriel Antonio; Sarmiento-Reyes, Arturo; Jimenez-Fernandez, Victor Manuel; Marin-Hernandez, Antonio; Herrera-May, Agustin Leobardo; Diaz-Sanchez, Alejandro; Huerta-Chua, Jesus

2014-01-01

In this article, we propose the application of a modified Taylor series method (MTSM) for the approximation of nonlinear problems described on finite intervals. The issue of Taylor series method with mixed boundary conditions is circumvented using shooting constants and extra derivatives of the problem. In order to show the benefits of this proposal, three different kinds of problems are solved: three-point boundary valued problem (BVP) of third-order with a hyperbolic sine nonlinearity, two-point BVP for a second-order nonlinear differential equation with an exponential nonlinearity, and a two-point BVP for a third-order nonlinear differential equation with a radical nonlinearity. The result shows that the MTSM method is capable to generate easily computable and highly accurate approximations for nonlinear equations. 34L30.

Case-Deletion Diagnostics for Nonlinear Structural Equation Models

ERIC Educational Resources Information Center

Lee, Sik-Yum; Lu, Bin

2003-01-01

In this article, a case-deletion procedure is proposed to detect influential observations in a nonlinear structural equation model. The key idea is to develop the diagnostic measures based on the conditional expectation of the complete-data log-likelihood function in the EM algorithm. An one-step pseudo approximation is proposed to reduce the…

One-Dimensional Fokker-Planck Equation with Quadratically Nonlinear Quasilocal Drift

NASA Astrophysics Data System (ADS)

Shapovalov, A. V.

2018-04-01

The Fokker-Planck equation in one-dimensional spacetime with quadratically nonlinear nonlocal drift in the quasilocal approximation is reduced with the help of scaling of the coordinates and time to a partial differential equation with a third derivative in the spatial variable. Determining equations for the symmetries of the reduced equation are derived and the Lie symmetries are found. A group invariant solution having the form of a traveling wave is found. Within the framework of Adomian's iterative method, the first iterations of an approximate solution of the Cauchy problem are obtained. Two illustrative examples of exact solutions are found.

Infinite hierarchy of nonlinear Schrödinger equations and their solutions.

PubMed

Ankiewicz, A; Kedziora, D J; Chowdury, A; Bandelow, U; Akhmediev, N

2016-01-01

We study the infinite integrable nonlinear Schrödinger equation hierarchy beyond the Lakshmanan-Porsezian-Daniel equation which is a particular (fourth-order) case of the hierarchy. In particular, we present the generalized Lax pair and generalized soliton solutions, plane wave solutions, Akhmediev breathers, Kuznetsov-Ma breathers, periodic solutions, and rogue wave solutions for this infinite-order hierarchy. We find that "even- order" equations in the set affect phase and "stretching factors" in the solutions, while "odd-order" equations affect the velocities. Hence odd-order equation solutions can be real functions, while even-order equation solutions are always complex.

Operational Solution to the Nonlinear Klein-Gordon Equation

NASA Astrophysics Data System (ADS)

Bengochea, G.; Verde-Star, L.; Ortigueira, M.

2018-05-01

We obtain solutions of the nonlinear Klein-Gordon equation using a novel operational method combined with the Adomian polynomial expansion of nonlinear functions. Our operational method does not use any integral transforms nor integration processes. We illustrate the application of our method by solving several examples and present numerical results that show the accuracy of the truncated series approximations to the solutions. Supported by Grant SEP-CONACYT 220603, the first author was supported by SEP-PRODEP through the project UAM-PTC-630, the third author was supported by Portuguese National Funds through the FCT Foundation for Science and Technology under the project PEst-UID/EEA/00066/2013

High-accuracy power series solutions with arbitrarily large radius of convergence for the fractional nonlinear Schrödinger-type equations

NASA Astrophysics Data System (ADS)

Khawaja, U. Al; Al-Refai, M.; Shchedrin, Gavriil; Carr, Lincoln D.

2018-06-01

Fractional nonlinear differential equations present an interplay between two common and important effective descriptions used to simplify high dimensional or more complicated theories: nonlinearity and fractional derivatives. These effective descriptions thus appear commonly in physical and mathematical modeling. We present a new series method providing systematic controlled accuracy for solutions of fractional nonlinear differential equations, including the fractional nonlinear Schrödinger equation and the fractional nonlinear diffusion equation. The method relies on spatially iterative use of power series expansions. Our approach permits an arbitrarily large radius of convergence and thus solves the typical divergence problem endemic to power series approaches. In the specific case of the fractional nonlinear Schrödinger equation we find fractional generalizations of cnoidal waves of Jacobi elliptic functions as well as a fractional bright soliton. For the fractional nonlinear diffusion equation we find the combination of fractional and nonlinear effects results in a more strongly localized solution which nevertheless still exhibits power law tails, albeit at a much lower density.

Nonlinear stability of the 1D Boltzmann equation in a periodic box

NASA Astrophysics Data System (ADS)

Wu, Kung-Chien

2018-05-01

We study the nonlinear stability of the Boltzmann equation in the 1D periodic box with size , where is the Knudsen number. The convergence rate is for small time region and exponential for large time region. Moreover, the exponential rate depends on the size of the domain (Knudsen number). This problem is highly nonlinear and hence we need more careful analysis to control the nonlinear term.

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

PubMed

Prieur, Fabrice; Vilenskiy, Gregory; Holm, Sverre

2012-10-01

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

Fast Neural Solution Of A Nonlinear Wave Equation

NASA Technical Reports Server (NTRS)

Barhen, Jacob; Toomarian, Nikzad

1996-01-01

Neural algorithm for simulation of class of nonlinear wave phenomena devised. Numerically solves special one-dimensional case of Korteweg-deVries equation. Intended to be executed rapidly by neural network implemented as charge-coupled-device/charge-injection device, very-large-scale integrated-circuit analog data processor of type described in "CCD/CID Processors Would Offer Greater Precision" (NPO-18972).

Extremely Fast Numerical Integration of Ocean Surface Wave Dynamics: Building Blocks for a Higher Order Method

DTIC Science & Technology

2006-09-30

equation known as the Kadomtsev - Petviashvili (KP) equation ): (ηt + coηx +αηηx + βη )x +γηyy = 0 (4) where γ = co / 2 . The KdV equation ...using the spectral formulation of the Kadomtsev - Petviashvili equation , a standard equation for nonlinear, shallow water wave dynamics that is a... Petviashvili and nonlinear Schroedinger equations and higher order corrections have been developed as prerequisites to coding the Boussinesq and Euler

A new Newton-like method for solving nonlinear equations.

PubMed

Saheya, B; Chen, Guo-Qing; Sui, Yun-Kang; Wu, Cai-Ying

2016-01-01

This paper presents an iterative scheme for solving nonline ar equations. We establish a new rational approximation model with linear numerator and denominator which has generalizes the local linear model. We then employ the new approximation for nonlinear equations and propose an improved Newton's method to solve it. The new method revises the Jacobian matrix by a rank one matrix each iteration and obtains the quadratic convergence property. The numerical performance and comparison show that the proposed method is efficient.

Solving nonlinear equilibrium equations of deformable systems by method of embedded polygons

NASA Astrophysics Data System (ADS)

Razdolsky, A. G.

2017-09-01

Solving of nonlinear algebraic equations is an obligatory stage of studying the equilibrium paths of nonlinear deformable systems. The iterative method for solving a system of nonlinear algebraic equations stated in an explicit or implicit form is developed in the present work. The method consists of constructing a sequence of polygons in Euclidean space that converge into a single point that displays the solution of the system. Polygon vertices are determined on the assumption that individual equations of the system are independent from each other and each of them is a function of only one variable. Initial positions of vertices for each subsequent polygon are specified at the midpoints of certain straight segments determined at the previous iteration. The present algorithm is applied for analytical investigation of the behavior of biaxially compressed nonlinear-elastic beam-column with an open thin-walled cross-section. Numerical examples are made for the I-beam-column on the assumption that its material follows a bilinear stress-strain diagram. A computer program based on the shooting method is developed for solving the problem. The method is reduced to numerical integration of a system of differential equations and to the solution of a system of nonlinear algebraic equations between the boundary values of displacements at the ends of the beam-column. A stress distribution at the beam-column cross-sections is determined by subdividing the cross-section area into many small cells. The equilibrium path for the twisting angle and the lateral displacements tend to the stationary point when the load is increased. Configuration of the path curves reveals that the ultimate load is reached shortly once the maximal normal stresses at the beam-column fall outside the limit of the elastic region. The beam-column has a unique equilibrium state for each value of the load, that is, there are no equilibrium states once the maximum load is reached.

Equations for description of nonlinear standing waves in constant-cross-sectioned resonators.

PubMed

Bednarik, Michal; Cervenka, Milan

2014-03-01

This work is focused on investigation of applicability of two widely used model equations for description of nonlinear standing waves in constant-cross-sectioned resonators. The investigation is based on the comparison of numerical solutions of these model equations with solutions of more accurate model equations whose validity has been verified experimentally in a number of published papers.

Traveling wave solution of driven nonlinear Schrödinger equation

NASA Astrophysics Data System (ADS)

Akbari-Moghanjoughi, M.

2017-09-01

The traveling solitary and cnoidal wave solutions of the one dimensional driven nonlinear Schrödinger equation with a generalized form of nonlinearity are presented in this paper. We examine the modulation of nonlinear solitary excitations in two known weakly nonlinear models of classic oscillators, namely, the Helmholtz and Duffing oscillators and envelope structure formations for different oscillator and driver parameters. It is shown that two distinct regimes of subcritical and supercritical modulations may occur for nonlinear excitations with propagation speeds v <√{4 F0 } and v >√{4 F0 } , respectively, in which F0 is the driver force strength. The envelope soliton and cnoidal waves in these regimes are observed to be fundamentally different. The effect of pseudoenergy on the structure of the modulated envelope excitations is studied in detail for both sub- and supercritical modulation types. The current model for traveling envelope excitations may be easily extended to pseudopotentials with full nonlinearity relevant to more realistic gases, fluids, and plasmas.

Derivation of nonlinear wave equations for ultrasound beam in nonuniform bubbly liquids

NASA Astrophysics Data System (ADS)

Kanagawa, Tetsuya; Yano, Takeru; Kawahara, Junya; Kobayashi, Kazumichi; Watanabe, Masao; Fujikawa, Shigeo

2012-09-01

Weakly nonlinear propagation of diffracted ultrasound beams in a nonuniform bubbly liquid is theoretically studied based on the method of multiple scales with the set of scaling relations of some physical parameters. It is assumed that the spatial distribution of the number density of bubbles in an initial state at rest is a slowly varying function of space coordinates and the amplitude of its variation is small compared with a mean number density. As a result, a Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation with dispersion and nonuniform effects for a low frequency case and a nonlinear Schrödinger (NLS) equation with dissipation, diffraction, and nonuniform effects for a high frequency case, are derived from the basic equations of bubbly flows.

Nonlinear inhomogeneous Fokker-Planck equations: Entropy and free-energy time evolution.

PubMed

Sicuro, Gabriele; Rapčan, Peter; Tsallis, Constantino

2016-12-01

We extend a recently introduced free-energy formalism for homogeneous Fokker-Planck equations to a wide, and physically appealing, class of inhomogeneous nonlinear Fokker-Planck equations. In our approach, the free-energy functional is expressed in terms of an entropic functional and an auxiliary potential, both derived from the coefficients of the equation. With reference to the introduced entropic functional, we discuss the entropy production in a relaxation process towards equilibrium. The properties of the stationary solutions of the considered Fokker-Planck equations are also discussed.

Closed form solutions of two time fractional nonlinear wave equations

NASA Astrophysics Data System (ADS)

Akbar, M. Ali; Ali, Norhashidah Hj. Mohd.; Roy, Ripan

2018-06-01

In this article, we investigate the exact traveling wave solutions of two nonlinear time fractional wave equations. The fractional derivatives are described in the sense of conformable fractional derivatives. In addition, the traveling wave solutions are accomplished in the form of hyperbolic, trigonometric, and rational functions involving free parameters. To investigate such types of solutions, we implement the new generalized (G‧ / G) -expansion method. The extracted solutions are reliable, useful and suitable to comprehend the optimal control problems, chaotic vibrations, global and local bifurcations and resonances, furthermore, fission and fusion phenomena occur in solitons, the relativistic energy-momentum relation, scalar electrodynamics, quantum relativistic one-particle theory, electromagnetic interactions etc. The results reveal that the method is very fruitful and convenient for exploring nonlinear differential equations of fractional order treated in theoretical physics.

Estimation of delays and other parameters in nonlinear functional differential equations

NASA Technical Reports Server (NTRS)

Banks, H. T.; Lamm, P. K. D.

1983-01-01

A spline-based approximation scheme for nonlinear nonautonomous delay differential equations is discussed. Convergence results (using dissipative type estimates on the underlying nonlinear operators) are given in the context of parameter estimation problems which include estimation of multiple delays and initial data as well as the usual coefficient-type parameters. A brief summary of some of the related numerical findings is also given.

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

NASA Technical Reports Server (NTRS)

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

2001-01-01

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

A lattice Boltzmann model with an amending function for simulating nonlinear partial differential equations

NASA Astrophysics Data System (ADS)

Chen, Lin-Jie; Ma, Chang-Feng

2010-01-01

This paper proposes a lattice Boltzmann model with an amending function for one-dimensional nonlinear partial differential equations (NPDEs) in the form ut + αuux + βunux + γuxx + δuxxx + ζuxxxx = 0. This model is different from existing models because it lets the time step be equivalent to the square of the space step and derives higher accuracy and nonlinear terms in NPDEs. With the Chapman-Enskog expansion, the governing evolution equation is recovered correctly from the continuous Boltzmann equation. The numerical results agree well with the analytical solutions.

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.

Nonzero solutions of nonlinear integral equations modeling infectious disease

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

Williams, L.R.; Leggett, R.W.

1982-01-01

Sufficient conditions to insure the existence of periodic solutions to the nonlinear integral equation, x(t) = ..integral../sup t//sub t-tau/f(s,x(s))ds, are given in terms of simple product and product integral inequalities. The equation can be interpreted as a model for the spread of infectious diseases (e.g., gonorrhea or any of the rhinovirus viruses) if x(t) is the proportion of infectives at time t and f(t,x(t)) is the proportion of new infectives per unit time.

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.

Direct application of Padé approximant for solving nonlinear differential equations.

PubMed

Vazquez-Leal, Hector; Benhammouda, Brahim; Filobello-Nino, Uriel; Sarmiento-Reyes, Arturo; Jimenez-Fernandez, Victor Manuel; Garcia-Gervacio, Jose Luis; Huerta-Chua, Jesus; Morales-Mendoza, Luis Javier; Gonzalez-Lee, Mario

2014-01-01

This work presents a direct procedure to apply Padé method to find approximate solutions for nonlinear differential equations. Moreover, we present some cases study showing the strength of the method to generate highly accurate rational approximate solutions compared to other semi-analytical methods. The type of tested nonlinear equations are: a highly nonlinear boundary value problem, a differential-algebraic oscillator problem, and an asymptotic problem. The high accurate handy approximations obtained by the direct application of Padé method shows the high potential if the proposed scheme to approximate a wide variety of problems. What is more, the direct application of the Padé approximant aids to avoid the previous application of an approximative method like Taylor series method, homotopy perturbation method, Adomian Decomposition method, homotopy analysis method, variational iteration method, among others, as tools to obtain a power series solutions to post-treat with the Padé approximant. 34L30.

Nonlinear equations of motion for the elastic bending and torsion of twisted nonuniform rotor blades

NASA Technical Reports Server (NTRS)

Hodges, D. H.; Dowell, E. H.

1974-01-01

The equations of motion are developed by two complementary methods, Hamilton's principle and the Newtonian method. The resulting equations are valid to second order for long, straight, slender, homogeneous, isotropic beams undergoing moderate displacements. The ordering scheme is based on the restriction that squares of the bending slopes, the torsion deformation, and the chord/radius and thickness/radius ratios are negligible with respect to unity. All remaining nonlinear terms are retained. The equations are valid for beams with mass centroid axis and area centroid (tension) axis offsets from the elastic axis, nonuniform mass and stiffness section properties, variable pretwist, and a small precone angle. The strain-displacement relations are developed from an exact transformation between the deformed and undeformed coordinate systems. These nonlinear relations form an important contribution to the final equations. Several nonlinear structural and inertial terms in the final equations are identified that can substantially influence the aeroelastic stability and response of hingeless helicopter rotor blades.

Topics Associated with Nonlinear Evolution Equations and Inverse Scattering in Multidimensions,

DTIC Science & Technology

1987-03-01

significant that these concepts can be generalized to 2 spatial plus one time dimension. Here the prototype equation is the Kadomtsev - Petviashvili (K-P...O-193 32 ? T TOPICS ASSOCIATED WITH NONLINEAR E VOLUTION EQUATIONS / AND INVERSE SCATTER! .(U) CLARKSON UNIV POTSDAM NY INST...8217 - Evolution Equations and L Inverse Scattering in Multi- dimensions by _i A ,’I Mark J. Ablowi ClrsnUiest PosaNwYr/37 LaRMFOMON* .F-5 Anwo~~~d kr /ua

Approximate analytic solutions to coupled nonlinear Dirac equations

DOE PAGES

Khare, Avinash; Cooper, Fred; Saxena, Avadh

2017-01-30

Here, we consider the coupled nonlinear Dirac equations (NLDEs) in 1+11+1 dimensions with scalar–scalar self-interactions g 1 2/2(more » $$\\bar{ψ}$$ψ) 2 + g 2 2/2($$\\bar{Φ}$$Φ) 2 + g 2 3($$\\bar{ψ}$$ψ)($$\\bar{Φ}$$Φ) as well as vector–vector interactions g 1 2/2($$\\bar{ψ}$$γμψ)($$\\bar{ψ}$$γμψ) + g 2 2/2($$\\bar{Φ}$$γμΦ)($$\\bar{Φ}$$γμΦ) + g 2 3($$\\bar{ψ}$$γμψ)($$\\bar{Φ}$$γμΦ). Writing the two components of the assumed rest frame solution of the coupled NLDE equations in the form ψ=e –iω1tR 1cosθ,R 1sinθΦ=e –iω2tR 2cosη,R 2sinη, and assuming that θ(x),η(x) have the same functional form they had when g3 = 0, which is an approximation consistent with the conservation laws, we then find approximate analytic solutions for Ri(x) which are valid for small values of g 3 2/g 2 2 and g 3 2/g 1 2. In the nonrelativistic limit we show that both of these coupled models go over to the same coupled nonlinear Schrödinger equation for which we obtain two exact pulse solutions vanishing at x → ±∞.« less

Approximate analytic solutions to coupled nonlinear Dirac equations

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

Khare, Avinash; Cooper, Fred; Saxena, Avadh

Here, we consider the coupled nonlinear Dirac equations (NLDEs) in 1+11+1 dimensions with scalar–scalar self-interactions g 1 2/2(more » $$\\bar{ψ}$$ψ) 2 + g 2 2/2($$\\bar{Φ}$$Φ) 2 + g 2 3($$\\bar{ψ}$$ψ)($$\\bar{Φ}$$Φ) as well as vector–vector interactions g 1 2/2($$\\bar{ψ}$$γμψ)($$\\bar{ψ}$$γμψ) + g 2 2/2($$\\bar{Φ}$$γμΦ)($$\\bar{Φ}$$γμΦ) + g 2 3($$\\bar{ψ}$$γμψ)($$\\bar{Φ}$$γμΦ). Writing the two components of the assumed rest frame solution of the coupled NLDE equations in the form ψ=e –iω1tR 1cosθ,R 1sinθΦ=e –iω2tR 2cosη,R 2sinη, and assuming that θ(x),η(x) have the same functional form they had when g3 = 0, which is an approximation consistent with the conservation laws, we then find approximate analytic solutions for Ri(x) which are valid for small values of g 3 2/g 2 2 and g 3 2/g 1 2. In the nonrelativistic limit we show that both of these coupled models go over to the same coupled nonlinear Schrödinger equation for which we obtain two exact pulse solutions vanishing at x → ±∞.« less

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.

Note on Solutions to a Class of Nonlinear Singular Integro-Differential Equations,

DTIC Science & Technology

1986-08-01

KdV) ut + 2uu x +Uxx x a 0, (1) the sine-Gordon equation Uxt a sin u, (2) and the Kadomtsev - Petviashvili (KP) equation (Ut + 2uu x + UXXx)x -3a 2u yy...SOUIN OA LSFNN ! /" / M.. \\boiz A.S ::-:- and ,M.O.. .- :1/1 / NOTE ON SOLUTIONS TO A CLASS OF NON \\ / LINEAR SINGULAR INTEGRO-DIFFERENTIA[ EQUATIONS by...important nonlinear evolution equations which can be linearized. Many of these equations fall into the category of linearization via soliton theory and

Vibrational Schroedinger Cats

NASA Technical Reports Server (NTRS)

Kis, Z.; Janszky, J.; Vinogradov, An. V.; Kobayashi, T.

1996-01-01

The optical Schroedinger cat states are simple realizations of quantum states having nonclassical features. It is shown that vibrational analogues of such states can be realized in an experiment of double pulse excitation of vibrionic transitions. To track the evolution of the vibrational wave packet we derive a non-unitary time evolution operator so that calculations are made in a quasi Heisenberg picture.

Finite-dimensional linear approximations of solutions to general irregular nonlinear operator equations and equations with quadratic operators

NASA Astrophysics Data System (ADS)

Kokurin, M. Yu.

2010-11-01

A general scheme for improving approximate solutions to irregular nonlinear operator equations in Hilbert spaces is proposed and analyzed in the presence of errors. A modification of this scheme designed for equations with quadratic operators is also examined. The technique of universal linear approximations of irregular equations is combined with the projection onto finite-dimensional subspaces of a special form. It is shown that, for finite-dimensional quadratic problems, the proposed scheme provides information about the global geometric properties of the intersections of quadrics.

Convolutionless Nakajima-Zwanzig equations for stochastic analysis in nonlinear dynamical systems.

PubMed

Venturi, D; Karniadakis, G E

2014-06-08

Determining the statistical properties of stochastic nonlinear systems is of major interest across many disciplines. Currently, there are no general efficient methods to deal with this challenging problem that involves high dimensionality, low regularity and random frequencies. We propose a framework for stochastic analysis in nonlinear dynamical systems based on goal-oriented probability density function (PDF) methods. The key idea stems from techniques of irreversible statistical mechanics, and it relies on deriving evolution equations for the PDF of quantities of interest, e.g. functionals of the solution to systems of stochastic ordinary and partial differential equations. Such quantities could be low-dimensional objects in infinite dimensional phase spaces. We develop the goal-oriented PDF method in the context of the time-convolutionless Nakajima-Zwanzig-Mori formalism. We address the question of approximation of reduced-order density equations by multi-level coarse graining, perturbation series and operator cumulant resummation. Numerical examples are presented for stochastic resonance and stochastic advection-reaction problems.

Convolutionless Nakajima–Zwanzig equations for stochastic analysis in nonlinear dynamical systems

PubMed Central

Venturi, D.; Karniadakis, G. E.

2014-01-01

Determining the statistical properties of stochastic nonlinear systems is of major interest across many disciplines. Currently, there are no general efficient methods to deal with this challenging problem that involves high dimensionality, low regularity and random frequencies. We propose a framework for stochastic analysis in nonlinear dynamical systems based on goal-oriented probability density function (PDF) methods. The key idea stems from techniques of irreversible statistical mechanics, and it relies on deriving evolution equations for the PDF of quantities of interest, e.g. functionals of the solution to systems of stochastic ordinary and partial differential equations. Such quantities could be low-dimensional objects in infinite dimensional phase spaces. We develop the goal-oriented PDF method in the context of the time-convolutionless Nakajima–Zwanzig–Mori formalism. We address the question of approximation of reduced-order density equations by multi-level coarse graining, perturbation series and operator cumulant resummation. Numerical examples are presented for stochastic resonance and stochastic advection–reaction problems. PMID:24910519

A three operator split-step method covering a larger set of non-linear partial differential equations

NASA Astrophysics Data System (ADS)

Zia, Haider

2017-06-01

This paper describes an updated exponential Fourier based split-step method that can be applied to a greater class of partial differential equations than previous methods would allow. These equations arise in physics and engineering, a notable example being the generalized derivative non-linear Schrödinger equation that arises in non-linear optics with self-steepening terms. These differential equations feature terms that were previously inaccessible to model accurately with low computational resources. The new method maintains a 3rd order error even with these additional terms and models the equation in all three spatial dimensions and time. The class of non-linear differential equations that this method applies to is shown. The method is fully derived and implementation of the method in the split-step architecture is shown. This paper lays the mathematical ground work for an upcoming paper employing this method in white-light generation simulations in bulk material.

Poincaré-MacMillan Equations of Motion for a Nonlinear Nonholonomic Dynamical System

NASA Astrophysics Data System (ADS)

Amjad, Hussain; Syed Tauseef, Mohyud-Din; Ahmet, Yildirim

2012-03-01

MacMillan's equations are extended to Poincaré's formalism, and MacMillan's equations for nonlinear nonholonomic systems are obtained in terms of Poincaré parameters. The equivalence of the results obtained here with other forms of equations of motion is demonstrated. An illustrative example of the theory is provided as well.

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

NASA Astrophysics Data System (ADS)

Szabo, Thomas L.

2003-10-01

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

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)

Generalized Appended Product Indicator Procedure for Nonlinear Structural Equation Analysis.

ERIC Educational Resources Information Center

Wall, Melanie M.; Amemiya, Yasuo

2001-01-01

Considers the estimation of polynomial structural models and shows a limitation of an existing method. Introduces a new procedure, the generalized appended product indicator procedure, for nonlinear structural equation analysis. Addresses statistical issues associated with the procedure through simulation. (SLD)

Nonlinear quantum Langevin equations for bosonic modes in solid-state systems

NASA Astrophysics Data System (ADS)

Manninen, Juuso; Agasti, Souvik; Massel, Francesco

2017-12-01

Based on the experimental evidence that impurities contribute to the dissipation properties of solid-state open quantum systems, we provide here a description in terms of nonlinear quantum Langevin equations of the role played by two-level systems in the dynamics of a bosonic degree of freedom. Our starting point is represented by the description of the system-environment coupling in terms of coupling to two separate reservoirs, modeling the interaction with external bosonic modes and two-level systems, respectively. Furthermore, we show how this model represents a specific example of a class of open quantum systems that can be described by nonlinear quantum Langevin equations. Our analysis offers a potential explanation of the parametric effects recently observed in circuit-QED cavity optomechanics experiments.

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

NASA Technical Reports Server (NTRS)

Gnoffo, Peter A.

2015-01-01

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

Khokhlov Zabolotskaya Kuznetsov type equation: nonlinear acoustics in heterogeneous media

NASA Astrophysics Data System (ADS)

Kostin, Ilya; Panasenko, Grigory

2006-04-01

The KZK type equation introduced in this Note differs from the traditional form of the KZK model known in acoustics by the assumptions on the nonlinear term. For this modified form, a global existence and uniqueness result is established for the case of non-constant coefficients. Afterwards the asymptotic behaviour of the solution of the KZK type equation with rapidly oscillating coefficients is studied. To cite this article: I. Kostin, G. Panasenko, C. R. Mecanique 334 (2006).

Quasi-periodic solutions of nonlinear beam equation with prescribed frequencies

NASA Astrophysics Data System (ADS)

Chang, Jing; Gao, Yixian; Li, Yong

2015-05-01

Consider the one dimensional nonlinear beam equation utt + uxxxx + mu + u3 = 0 under Dirichlet boundary conditions. We show that for any m > 0 but a set of small Lebesgue measure, the above equation admits a family of small-amplitude quasi-periodic solutions with n-dimensional Diophantine frequencies. These Diophantine frequencies are the small dilation of a prescribed Diophantine vector. The proofs are based on an infinite dimensional Kolmogorov-Arnold-Moser iteration procedure and a partial Birkhoff normal form.

Neuro-evolutionary computing paradigm for Painlevé equation-II in nonlinear optics

NASA Astrophysics Data System (ADS)

Ahmad, Iftikhar; Ahmad, Sufyan; Awais, Muhammad; Ul Islam Ahmad, Siraj; Asif Zahoor Raja, Muhammad

2018-05-01

The aim of this study is to investigate the numerical treatment of the Painlevé equation-II arising in physical models of nonlinear optics through artificial intelligence procedures by incorporating a single layer structure of neural networks optimized with genetic algorithms, sequential quadratic programming and active set techniques. We constructed a mathematical model for the nonlinear Painlevé equation-II with the help of networks by defining an error-based cost function in mean square sense. The performance of the proposed technique is validated through statistical analyses by means of the one-way ANOVA test conducted on a dataset generated by a large number of independent runs.

Study of coupled nonlinear partial differential equations for finding exact analytical solutions.

PubMed

Khan, Kamruzzaman; Akbar, M Ali; Koppelaar, H

2015-07-01

Exact solutions of nonlinear partial differential equations (NPDEs) are obtained via the enhanced (G'/G)-expansion method. The method is subsequently applied to find exact solutions of the Drinfel'd-Sokolov-Wilson (DSW) equation and the (2+1)-dimensional Painlevé integrable Burgers (PIB) equation. The efficiency of this method for finding these exact solutions is demonstrated. The method is effective and applicable for many other NPDEs in mathematical physics.

The study of nonlinear almost periodic differential equations without recourse to the H-classes of these equations

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

Slyusarchuk, V. E., E-mail: V.E.Slyusarchuk@gmail.com, E-mail: V.Ye.Slyusarchuk@NUWM.rv.ua

2014-06-01

The well-known theorems of Favard and Amerio on the existence of almost periodic solutions to linear and nonlinear almost periodic differential equations depend to a large extent on the H-classes and the requirement that the bounded solutions of these equations be separated. The present paper provides different conditions for the existence of almost periodic solutions. These conditions, which do not depend on the H-classes of the equations, are formulated in terms of a special functional on the set of bounded solutions of the equations under consideration. This functional is used, in particular, to test whether solutions are separated. Bibliography: 24more » titles. (paper)« less

The existence of periodic solutions for nonlinear beam equations on Td by a para-differential method

NASA Astrophysics Data System (ADS)

Chen, Bochao; Li, Yong; Gao, Yixian

2018-05-01

This paper focuses on the construction of periodic solutions of nonlinear beam equations on the $d$-dimensional tori. For a large set of frequencies, we demonstrate that an equivalent form of the nonlinear equations can be obtained by a para-differential conjugation. Given the non-resonant conditions on each finite dimensional subspaces, it is shown that the periodic solutions can be constructed for the block diagonal equation by a classical iteration scheme.

Non-linear analysis of wave progagation using transform methods and plates and shells using integral equations

NASA Astrophysics Data System (ADS)

Pipkins, Daniel Scott

Two diverse topics of relevance in modern computational mechanics are treated. The first involves the modeling of linear and non-linear wave propagation in flexible, lattice structures. The technique used combines the Laplace Transform with the Finite Element Method (FEM). The procedure is to transform the governing differential equations and boundary conditions into the transform domain where the FEM formulation is carried out. For linear problems, the transformed differential equations can be solved exactly, hence the method is exact. As a result, each member of the lattice structure is modeled using only one element. In the non-linear problem, the method is no longer exact. The approximation introduced is a spatial discretization of the transformed non-linear terms. The non-linear terms are represented in the transform domain by making use of the complex convolution theorem. A weak formulation of the resulting transformed non-linear equations yields a set of element level matrix equations. The trial and test functions used in the weak formulation correspond to the exact solution of the linear part of the transformed governing differential equation. Numerical results are presented for both linear and non-linear systems. The linear systems modeled are longitudinal and torsional rods and Bernoulli-Euler and Timoshenko beams. For non-linear systems, a viscoelastic rod and Von Karman type beam are modeled. The second topic is the analysis of plates and shallow shells under-going finite deflections by the Field/Boundary Element Method. Numerical results are presented for two plate problems. The first is the bifurcation problem associated with a square plate having free boundaries which is loaded by four, self equilibrating corner forces. The results are compared to two existing numerical solutions of the problem which differ substantially. non-linear model are compared to those

FAST TRACK COMMUNICATION: The nonlinear fragmentation equation

NASA Astrophysics Data System (ADS)

Ernst, Matthieu H.; Pagonabarraga, Ignacio

2007-04-01

We study the kinetics of nonlinear irreversible fragmentation. Here, fragmentation is induced by interactions/collisions between pairs of particles and modelled by general classes of interaction kernels, for several types of breakage models. We construct initial value and scaling solutions of the fragmentation equations, and apply the 'non-vanishing mass flux' criterion for the occurrence of shattering transitions. These properties enable us to determine the phase diagram for the occurrence of shattering states and of scaling states in the phase space of model parameters.

The hair-trigger effect for a class of nonlocal nonlinear equations

NASA Astrophysics Data System (ADS)

Finkelshtein, Dmitri; Tkachov, Pasha

2018-06-01

We prove the hair-trigger effect for a class of nonlocal nonlinear evolution equations on which have only two constant stationary solutions, 0 and . The effect consists in that the solution with an initial condition non identical to zero converges (when time goes to ) to θ locally uniformly in . We also find sufficient conditions for existence, uniqueness and comparison principle in the considered equations.

A nonlinear ordinary differential equation associated with the quantum sojourn time

NASA Astrophysics Data System (ADS)

Benguria, Rafael D.; Duclos, Pierre; Fernández, Claudio; Sing-Long, Carlos

2010-11-01

We study a nonlinear ordinary differential equation on the half-line, with the Dirichlet boundary condition at the origin. This equation arises when studying the local maxima of the sojourn time for a free quantum particle whose states belong to an adequate subspace of the unit sphere of the corresponding Hilbert space. We establish several results concerning the existence and asymptotic behavior of the solutions.

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

NASA Technical Reports Server (NTRS)

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

2002-01-01

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

Cavity equations for a positive- or negative-refraction-index material with electric and magnetic nonlinearities.

PubMed

Mártin, Daniel A; Hoyuelos, Miguel

2009-11-01

We study evolution equations for electric and magnetic field amplitudes in a ring cavity with plane mirrors. The cavity is filled with a positive or negative-refraction-index material with third-order effective electric and magnetic nonlinearities. Two coupled nonlinear equations for the electric and magnetic amplitudes are obtained. We prove that the description can be reduced to one Lugiato-Lefever equation with generalized coefficients. A stability analysis of the homogeneous solution, complemented with numerical integration, shows that any combination of the parameters should correspond to one of three characteristic behaviors.

Nonlinear Aeroelastic Equations of Motion of Twisted, Nonuniform, Flexible Horizontal-Axis Wind Turbine Blades

NASA Technical Reports Server (NTRS)

Kaza, K. R. V.

1980-01-01

The second-degree nonlinear equations of motion for a flexible, twisted, nonuniform, horizontal axis wind turbine blade were developed using Hamilton's principle. A mathematical ordering scheme which was consistent with the assumption of a slender beam was used to discard some higher-order elastic and inertial terms in the second-degree nonlinear equations. The blade aerodynamic loading which was employed accounted for both wind shear and tower shadow and was obtained from strip theory based on a quasi-steady approximation of two-dimensional, incompressible, unsteady, airfoil theory. The resulting equations had periodic coefficients and were suitable for determining the aeroelastic stability and response of large horizontal-axis wind turbine blades.

Nondimensional Parameters and Equations for Nonlinear and Bifurcation Analyses of Thin Anisotropic Quasi-Shallow Shells

NASA Technical Reports Server (NTRS)

Nemeth, Michael P.

2010-01-01

A comprehensive development of nondimensional parameters and equations for nonlinear and bifurcations analyses of quasi-shallow shells, based on the Donnell-Mushtari-Vlasov theory for thin anisotropic shells, is presented. A complete set of field equations for geometrically imperfect shells is presented in terms general of lines-of-curvature coordinates. A systematic nondimensionalization of these equations is developed, several new nondimensional parameters are defined, and a comprehensive stress-function formulation is presented that includes variational principles for equilibrium and compatibility. Bifurcation analysis is applied to the nondimensional nonlinear field equations and a comprehensive set of bifurcation equations are presented. An extensive collection of tables and figures are presented that show the effects of lamina material properties and stacking sequence on the nondimensional parameters.

A computationally efficient scheme for the non-linear diffusion equation

NASA Astrophysics Data System (ADS)

Termonia, P.; Van de Vyver, H.

2009-04-01

This Letter proposes a new numerical scheme for integrating the non-linear diffusion equation. It is shown that it is linearly stable. Some tests are presented comparing this scheme to a popular decentered version of the linearized Crank-Nicholson scheme, showing that, although this scheme is slightly less accurate in treating the highly resolved waves, (i) the new scheme better treats highly non-linear systems, (ii) better handles the short waves, (iii) for a given test bed turns out to be three to four times more computationally cheap, and (iv) is easier in implementation.

Study of coupled nonlinear partial differential equations for finding exact analytical solutions

PubMed Central

Khan, Kamruzzaman; Akbar, M. Ali; Koppelaar, H.

2015-01-01

Exact solutions of nonlinear partial differential equations (NPDEs) are obtained via the enhanced (G′/G)-expansion method. The method is subsequently applied to find exact solutions of the Drinfel'd–Sokolov–Wilson (DSW) equation and the (2+1)-dimensional Painlevé integrable Burgers (PIB) equation. The efficiency of this method for finding these exact solutions is demonstrated. The method is effective and applicable for many other NPDEs in mathematical physics. PMID:26587256

Averaging Principle for the Higher Order Nonlinear Schrödinger Equation with a Random Fast Oscillation

NASA Astrophysics Data System (ADS)

Gao, Peng

2018-04-01

This work concerns the problem associated with averaging principle for a higher order nonlinear Schrödinger equation perturbed by a oscillating term arising as the solution of a stochastic reaction-diffusion equation evolving with respect to the fast time. This model can be translated into a multiscale stochastic partial differential equations. Stochastic averaging principle is a powerful tool for studying qualitative analysis of stochastic dynamical systems with different time-scales. To be more precise, under suitable conditions, we prove that there is a limit process in which the fast varying process is averaged out and the limit process which takes the form of the higher order nonlinear Schrödinger equation is an average with respect to the stationary measure of the fast varying process. Finally, by using the Khasminskii technique we can obtain the rate of strong convergence for the slow component towards the solution of the averaged equation, and as a consequence, the system can be reduced to a single higher order nonlinear Schrödinger equation with a modified coefficient.

Averaging Principle for the Higher Order Nonlinear Schrödinger Equation with a Random Fast Oscillation

NASA Astrophysics Data System (ADS)

Gao, Peng

2018-06-01

This work concerns the problem associated with averaging principle for a higher order nonlinear Schrödinger equation perturbed by a oscillating term arising as the solution of a stochastic reaction-diffusion equation evolving with respect to the fast time. This model can be translated into a multiscale stochastic partial differential equations. Stochastic averaging principle is a powerful tool for studying qualitative analysis of stochastic dynamical systems with different time-scales. To be more precise, under suitable conditions, we prove that there is a limit process in which the fast varying process is averaged out and the limit process which takes the form of the higher order nonlinear Schrödinger equation is an average with respect to the stationary measure of the fast varying process. Finally, by using the Khasminskii technique we can obtain the rate of strong convergence for the slow component towards the solution of the averaged equation, and as a consequence, the system can be reduced to a single higher order nonlinear Schrödinger equation with a modified coefficient.

Exact soliton solutions and their stability control in the nonlinear Schrödinger equation with spatiotemporally modulated nonlinearity.

PubMed

Tian, Qing; Wu, Lei; Zhang, Jie-Fang; Malomed, Boris A; Mihalache, D; Liu, W M

2011-01-01

We put forward a generic transformation which helps to find exact soliton solutions of the nonlinear Schrödinger equation with a spatiotemporal modulation of the nonlinearity and external potentials. As an example, we construct exact solitons for the defocusing nonlinearity and harmonic potential. When the soliton's eigenvalue is fixed, the number of exact solutions is determined by energy levels of the linear harmonic oscillator. In addition to the stable fundamental solitons, stable higher-order modes, describing array of dark solitons nested in a finite-width background, are constructed too. We also show how to control the instability domain of the nonstationary solitons.

Khater method for nonlinear Sharma Tasso-Olever (STO) equation of fractional order

NASA Astrophysics Data System (ADS)

Bibi, Sadaf; Mohyud-Din, Syed Tauseef; Khan, Umar; Ahmed, Naveed

In this work, we have implemented a direct method, known as Khater method to establish exact solutions of nonlinear partial differential equations of fractional order. Number of solutions provided by this method is greater than other traditional methods. Exact solutions of nonlinear fractional order Sharma Tasso-Olever (STO) equation are expressed in terms of kink, travelling wave, periodic and solitary wave solutions. Modified Riemann-Liouville derivative and Fractional complex transform have been used for compatibility with fractional order sense. Solutions have been graphically simulated for understanding the physical aspects and importance of the method. A comparative discussion between our established results and the results obtained by existing ones is also presented. Our results clearly reveal that the proposed method is an effective, powerful and straightforward technique to work out new solutions of various types of differential equations of non-integer order in the fields of applied sciences and engineering.

Loss of Energy Concentration in Nonlinear Evolution Beam Equations

NASA Astrophysics Data System (ADS)

Garrione, Maurizio; Gazzola, Filippo

2017-12-01

Motivated by the oscillations that were seen at the Tacoma Narrows Bridge, we introduce the notion of solutions with a prevailing mode for the nonlinear evolution beam equation u_{tt} + u_{xxxx} + f(u)= g(x, t) in bounded space-time intervals. We give a new definition of instability for these particular solutions, based on the loss of energy concentration on their prevailing mode. We distinguish between two different forms of energy transfer, one physiological (unavoidable and depending on the nonlinearity) and one due to the insurgence of instability. We then prove a theoretical result allowing to reduce the study of this kind of infinite-dimensional stability to that of a finite-dimensional approximation. With this background, we study the occurrence of instability for three different kinds of nonlinearities f and for some forcing terms g, highlighting some of their structural properties and performing some numerical simulations.

From the nonlinear Fokker-Planck equation to the Vlasov description and back: Confined interacting particles with drag

NASA Astrophysics Data System (ADS)

Plastino, A. R.; Curado, E. M. F.; Nobre, F. D.; Tsallis, C.

2018-02-01

Nonlinear Fokker-Planck equations endowed with power-law diffusion terms have proven to be valuable tools for the study of diverse complex systems in physics, biology, and other fields. The nonlinearity appearing in these evolution equations can be interpreted as providing an effective description of a system of particles interacting via short-range forces while performing overdamped motion under the effect of an external confining potential. This point of view has been recently applied to the study of thermodynamical features of interacting vortices in type II superconductors. In the present work we explore an embedding of the nonlinear Fokker-Planck equation within a Vlasov equation, thus incorporating inertial effects to the concomitant particle dynamics. Exact time-dependent solutions of the q -Gaussian form (with compact support) are obtained for the Vlasov equation in the case of quadratic confining potentials.

Asymptotic stability of a nonlinear Korteweg-de Vries equation with critical lengths

NASA Astrophysics Data System (ADS)

Chu, Jixun; Coron, Jean-Michel; Shang, Peipei

2015-10-01

We study an initial-boundary-value problem of a nonlinear Korteweg-de Vries equation posed on the finite interval (0, 2 kπ) where k is a positive integer. The whole system has Dirichlet boundary condition at the left end-point, and both of Dirichlet and Neumann homogeneous boundary conditions at the right end-point. It is known that the origin is not asymptotically stable for the linearized system around the origin. We prove that the origin is (locally) asymptotically stable for the nonlinear system if the integer k is such that the kernel of the linear Korteweg-de Vries stationary equation is of dimension 1. This is for example the case if k = 1.

Stable dipole solitons and soliton complexes in the nonlinear Schrödinger equation with periodically modulated nonlinearity

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

Lebedev, M. E., E-mail: gloriouslair@gmail.com, E-mail: galfimov@yahoo.com; Alfimov, G. L., E-mail: gloriouslair@gmail.com, E-mail: galfimov@yahoo.com; Malomed, Boris A., E-mail: malomed@post.tau.ac.il

We develop a general classification of the infinite number of families of solitons and soliton complexes in the one-dimensional Gross-Pitaevskii/nonlinear Schrödinger equation with a nonlinear lattice pseudopotential, i.e., periodically modulated coefficient in front of the cubic term, which takes both positive and negative local values. This model finds direct implementations in atomic Bose-Einstein condensates and nonlinear optics. The most essential finding is the existence of two branches of dipole solitons (DSs), which feature an antisymmetric shape, being essentially squeezed into a single cell of the nonlinear lattice. This soliton species was not previously considered in nonlinear lattices. We demonstrate thatmore » one branch of the DS family (namely, which obeys the Vakhitov-Kolokolov criterion) is stable, while unstable DSs spontaneously transform into stable fundamental solitons (FSs). The results are obtained in numerical and approximate analytical forms, the latter based on the variational approximation. Some stable bound states of FSs are found too.« less

Via generalized function projective synchronization in nonlinear Schrödinger equation for secure communication

NASA Astrophysics Data System (ADS)

Zhao, L. W.; Du, J. G.; Yin, J. L.

2018-05-01

This paper proposes a novel secured communication scheme in a chaotic system by applying generalized function projective synchronization of the nonlinear Schrödinger equation. This phenomenal approach guarantees a secured and convenient communication. Our study applied the Melnikov theorem with an active control strategy to suppress chaos in the system. The transmitted information signal is modulated into the parameter of the nonlinear Schrödinger equation in the transmitter and it is assumed that the parameter of the receiver system is unknown. Based on the Lyapunov stability theory and the adaptive control technique, the controllers are designed to make two identical nonlinear Schrödinger equation with the unknown parameter asymptotically synchronized. The numerical simulation results of our study confirmed the validity, effectiveness and the feasibility of the proposed novel synchronization method and error estimate for a secure communication. The Chaos masking signals of the information communication scheme, further guaranteed a safer and secured information communicated via this approach.

Preconditioned steepest descent methods for some nonlinear elliptic equations involving p-Laplacian terms

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

Feng, Wenqiang, E-mail: wfeng1@vols.utk.edu; Salgado, Abner J., E-mail: asalgad1@utk.edu; Wang, Cheng, E-mail: cwang1@umassd.edu

We describe and analyze preconditioned steepest descent (PSD) solvers for fourth and sixth-order nonlinear elliptic equations that include p-Laplacian terms on periodic domains in 2 and 3 dimensions. The highest and lowest order terms of the equations are constant-coefficient, positive linear operators, which suggests a natural preconditioning strategy. Such nonlinear elliptic equations often arise from time discretization of parabolic equations that model various biological and physical phenomena, in particular, liquid crystals, thin film epitaxial growth and phase transformations. The analyses of the schemes involve the characterization of the strictly convex energies associated with the equations. We first give a generalmore » framework for PSD in Hilbert spaces. Based on certain reasonable assumptions of the linear pre-conditioner, a geometric convergence rate is shown for the nonlinear PSD iteration. We then apply the general theory to the fourth and sixth-order problems of interest, making use of Sobolev embedding and regularity results to confirm the appropriateness of our pre-conditioners for the regularized p-Lapacian problems. Our results include a sharper theoretical convergence result for p-Laplacian systems compared to what may be found in existing works. We demonstrate rigorously how to apply the theory in the finite dimensional setting using finite difference discretization methods. Numerical simulations for some important physical application problems – including thin film epitaxy with slope selection and the square phase field crystal model – are carried out to verify the efficiency of the scheme.« less

Preconditioned steepest descent methods for some nonlinear elliptic equations involving p-Laplacian terms

NASA Astrophysics Data System (ADS)

Feng, Wenqiang; Salgado, Abner J.; Wang, Cheng; Wise, Steven M.

2017-04-01

We describe and analyze preconditioned steepest descent (PSD) solvers for fourth and sixth-order nonlinear elliptic equations that include p-Laplacian terms on periodic domains in 2 and 3 dimensions. The highest and lowest order terms of the equations are constant-coefficient, positive linear operators, which suggests a natural preconditioning strategy. Such nonlinear elliptic equations often arise from time discretization of parabolic equations that model various biological and physical phenomena, in particular, liquid crystals, thin film epitaxial growth and phase transformations. The analyses of the schemes involve the characterization of the strictly convex energies associated with the equations. We first give a general framework for PSD in Hilbert spaces. Based on certain reasonable assumptions of the linear pre-conditioner, a geometric convergence rate is shown for the nonlinear PSD iteration. We then apply the general theory to the fourth and sixth-order problems of interest, making use of Sobolev embedding and regularity results to confirm the appropriateness of our pre-conditioners for the regularized p-Lapacian problems. Our results include a sharper theoretical convergence result for p-Laplacian systems compared to what may be found in existing works. We demonstrate rigorously how to apply the theory in the finite dimensional setting using finite difference discretization methods. Numerical simulations for some important physical application problems - including thin film epitaxy with slope selection and the square phase field crystal model - are carried out to verify the efficiency of the scheme.

Using Microcomputers to Teach Non-Linear Equations at Sixth Form Level.

ERIC Educational Resources Information Center

Cheung, Y. L.

1984-01-01

Promotes the use of the microcomputer in mathematics instruction, reviewing approaches to teaching nonlinear equations. Examples of computer diagrams are illustrated and compared to textbook samples. An example of a problem-solving program is included. (ML)

On the integration of a class of nonlinear systems of ordinary differential equations

NASA Astrophysics Data System (ADS)

Talyshev, Aleksandr A.

2017-11-01

For each associative, commutative, and unitary algebra over the field of real or complex numbers and an integrable nonlinear ordinary differential equation we can to construct integrable systems of ordinary differential equations and integrable systems of partial differential equations. In this paper we consider in some sense the inverse problem. Determine the conditions under which a given system of ordinary differential equations can be represented as a differential equation in some associative, commutative and unitary algebra. It is also shown that associativity is not a necessary condition.

A method of solving simple harmonic oscillator Schroedinger equation

NASA Technical Reports Server (NTRS)

Maury, Juan Carlos F.

1995-01-01

A usual step in solving totally Schrodinger equation is to try first the case when dimensionless position independent variable w is large. In this case the Harmonic Oscillator equation takes the form (d(exp 2)/dw(exp 2) - w(exp 2))F = 0, and following W.K.B. method, it gives the intermediate corresponding solution F = exp(-w(exp 2)/2), which actually satisfies exactly another equation, (d(exp 2)/dw(exp 2) + 1 - w(exp 2))F = 0. We apply a different method, useful in anharmonic oscillator equations, similar to that of Rampal and Datta, and although it is slightly more complicated however it is also more general and systematic.

Optical soliton solutions of the cubic-quintic non-linear Schrödinger's equation including an anti-cubic term

NASA Astrophysics Data System (ADS)

Kaplan, Melike; Hosseini, Kamyar; Samadani, Farzan; Raza, Nauman

2018-07-01

A wide range of problems in different fields of the applied sciences especially non-linear optics is described by non-linear Schrödinger's equations (NLSEs). In the present paper, a specific type of NLSEs known as the cubic-quintic non-linear Schrödinger's equation including an anti-cubic term has been studied. The generalized Kudryashov method along with symbolic computation package has been exerted to carry out this objective. As a consequence, a series of optical soliton solutions have formally been retrieved. It is corroborated that the generalized form of Kudryashov method is a direct, effectual, and reliable technique to deal with various types of non-linear Schrödinger's equations.

Lie symmetry analysis, explicit solutions and conservation laws for the space-time fractional nonlinear evolution equations

NASA Astrophysics Data System (ADS)

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

2018-04-01

This paper studies the symmetry analysis, explicit solutions, convergence analysis, and conservation laws (Cls) for two different space-time fractional nonlinear evolution equations with Riemann-Liouville (RL) derivative. The governing equations are reduced to nonlinear ordinary differential equation (ODE) of fractional order using their Lie point symmetries. In the reduced equations, the derivative is in Erdelyi-Kober (EK) sense, power series technique is applied to derive an explicit solutions for the reduced fractional ODEs. The convergence of the obtained power series solutions is also presented. Moreover, the new conservation theorem and the generalization of the Noether operators are developed to construct the nonlocal Cls for the equations . Some interesting figures for the obtained explicit solutions are presented.

Modulation of localized solutions in a system of two coupled nonlinear Schrödinger equations.

PubMed

Cardoso, W B; Avelar, A T; Bazeia, D

2012-08-01

In this work we study localized solutions of a system of two coupled nonlinear Schrödinger equations, with the linear (potential) and nonlinear coefficients engendering spatial and temporal dependencies. Similarity transformations are used to convert the nonautonomous coupled equations into autonomous ones and we use the trial orbit method to help us solving them, presenting solutions in a general way. Numerical experiments are then used to verify the stability of the localized solutions.

Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities

PubMed Central

Vázquez, J. L.

2010-01-01

The goal of this paper is to state the optimal decay rate for solutions of the nonlinear fast diffusion equation and, in self-similar variables, the optimal convergence rates to Barenblatt self-similar profiles and their generalizations. It relies on the identification of the optimal constants in some related Hardy–Poincaré inequalities and concludes a long series of papers devoted to generalized entropies, functional inequalities, and rates for nonlinear diffusion equations. PMID:20823259

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

NASA Astrophysics Data System (ADS)

Vrecica, Teodor; Toledo, Yaron

2015-04-01

One-dimensional deterministic and stochastic evolution equations are derived for the dispersive nonlinear waves while taking dissipation of energy into account. The deterministic nonlinear evolution equations are formulated using operational calculus by following the approach of Bredmose et al. (2005). Their formulation is extended to include the linear and nonlinear effects of wave dissipation due to friction and breaking. The resulting equation set describes the linear evolution of the velocity potential for each wave harmonic coupled by quadratic nonlinear terms. These terms describe the nonlinear interactions between triads of waves, which represent the leading-order nonlinear effects in the near-shore region. The equations are translated to the amplitudes of the surface elevation by using the approach of Agnon and Sheremet (1997) with the correction of Eldeberky and Madsen (1999). The only current possibility for calculating the surface gravity wave field over large domains is by using stochastic wave evolution models. Hence, the above deterministic model is formulated as a stochastic one using the method of Agnon and Sheremet (1997) with two types of stochastic closure relations (Benney and Saffman's, 1966, and Hollway's, 1980). These formulations cannot be applied to the common wave forecasting models without further manipulation, as they include a non-local wave shoaling coefficients (i.e., ones that require integration along the wave rays). Therefore, a localization method was applied (see Stiassnie and Drimer, 2006, and Toledo and Agnon, 2012). This process essentially extracts the local terms that constitute the mean nonlinear energy transfer while discarding the remaining oscillatory terms, which transfer energy back and forth. One of the main findings of this work is the understanding that the approximated non-local coefficients behave in two essentially different manners. In intermediate water depths these coefficients indeed consist of rapidly

A Nonlinear Diffusion Equation-Based Model for Ultrasound Speckle Noise Removal

NASA Astrophysics Data System (ADS)

Zhou, Zhenyu; Guo, Zhichang; Zhang, Dazhi; Wu, Boying

2018-04-01

Ultrasound images are contaminated by speckle noise, which brings difficulties in further image analysis and clinical diagnosis. In this paper, we address this problem in the view of nonlinear diffusion equation theories. We develop a nonlinear diffusion equation-based model by taking into account not only the gradient information of the image, but also the information of the gray levels of the image. By utilizing the region indicator as the variable exponent, we can adaptively control the diffusion type which alternates between the Perona-Malik diffusion and the Charbonnier diffusion according to the image gray levels. Furthermore, we analyze the proposed model with respect to the theoretical and numerical properties. Experiments show that the proposed method achieves much better speckle suppression and edge preservation when compared with the traditional despeckling methods, especially in the low gray level and low-contrast regions.

Effective equations for the quantum pendulum from momentous quantum mechanics

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

Hernandez, Hector H.; Chacon-Acosta, Guillermo; Departamento de Matematicas Aplicadas y Sistemas, Universidad Autonoma Metropolitana-Cuajimalpa, Artificios 40, Mexico D. F. 01120

In this work we study the quantum pendulum within the framework of momentous quantum mechanics. This description replaces the Schroedinger equation for the quantum evolution of the system with an infinite set of classical equations for expectation values of configuration variables, and quantum dispersions. We solve numerically the effective equations up to the second order, and describe its evolution.

Spinodal Decomposition for theCahn-Hilliard Equation in Higher Dimensions:Nonlinear Dynamics

NASA Astrophysics Data System (ADS)

Maier-Paape, Stanislaus; Wanner, Thomas

This paper addresses the phenomenon of spinodal decomposition for the Cahn-Hilliard equation where Ω⊂n, n∈{1,2,3 }, is a bounded domain with sufficiently smooth boundary, and f is cubic-like, for example f(u) =u-u3. Based on the results of [26] the nonlinear Cahn-Hilliard equation will be discussed. This equation generates a nonlinear semiflow in certain affine subspaces of H2(Ω). In a neighborhood Uɛ with size proportional to ɛn around the constant solution , where μ lies in the spinodal region, we observe the following behavior. Within a local inertial manifold containing there exists a finite-dimensional invariant manifold which dominates the behavior of all solutions starting with initial conditions from a small ball around with probability almost 1. The dimension of is proportional to ɛ-n and the elements of exhibit a common geometric quantity which is strongly related to a characteristic wavelength proportional to ɛ.

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.

Finite elements and fluid dynamics. [instability effects on solution of nonlinear equations

NASA Technical Reports Server (NTRS)

Fix, G.

1975-01-01

Difficulties concerning a use of the finite element method in the solution of the nonlinear equations of fluid dynamics are partly related to various 'hidden' instabilities which often arise in fluid calculations. The instabilities are typically due to boundary effects or nonlinearities. It is shown that in certain cases these instabilities can be avoided if certain conservation laws are satisfied, and that the latter are often intimately related to finite elements.

Maximal amplitudes of finite-gap solutions for the focusing Nonlinear Schrödinger Equation

NASA Astrophysics Data System (ADS)

Bertola, M.; Tovbis, A.

2017-09-01

Finite-gap (algebro-geometric) solutions to the focusing Nonlinear Schrödinger Equation (fNLS) i ψ_t + ψ_{xx} + 2|ψ|^2ψ=0, are quasi-periodic solutions that represent nonlinear multi-phase waves. In general, a finite-gap solution for (0-1) is defined by a collection of Schwarz symmetrical spectral bands and of real constants (initial phases), associated with the corresponding bands. In this paper we prove an interesting new formula for the maximal amplitude of a finite-gap solution to the focusing Nonlinear Schrödinger equation with given spectral bands: the amplitude does not exceed the sum of the imaginary parts of all the endpoints in the upper half plane. In the case of the straight vertical bands, that amounts to the half of the sum of the length of all the bands. The maximal amplitude will be attained for certain choices of the initial phases. This result is an important part of a criterion for the potential presence of the rogue waves in finite-gap solutions with a given set of spectral endpoints, obtained in Bertola et al. (Proc R Soc A, 2016. doi: 10.1098/rspa.2016.0340). A similar result was also obtained for the defocusing Nonlinear Schrödinger equation.

Reduced-order modelling of parameter-dependent, linear and nonlinear dynamic partial differential equation models.

PubMed

Shah, A A; Xing, W W; Triantafyllidis, V

2017-04-01

In this paper, we develop reduced-order models for dynamic, parameter-dependent, linear and nonlinear partial differential equations using proper orthogonal decomposition (POD). The main challenges are to accurately and efficiently approximate the POD bases for new parameter values and, in the case of nonlinear problems, to efficiently handle the nonlinear terms. We use a Bayesian nonlinear regression approach to learn the snapshots of the solutions and the nonlinearities for new parameter values. Computational efficiency is ensured by using manifold learning to perform the emulation in a low-dimensional space. The accuracy of the method is demonstrated on a linear and a nonlinear example, with comparisons with a global basis approach.

Bright-dark and dark-dark solitons for the coupled cubic-quintic nonlinear Schrödinger equations in a twin-core nonlinear optical fiber

NASA Astrophysics Data System (ADS)

Yuan, Yu-Qiang; Tian, Bo; Liu, Lei; Chai, Han-Peng

2017-11-01

In this paper, we investigate the coupled cubic-quintic nonlinear Schrödinger equations, which can describe the effects of quintic nonlinearity on the ultrashort optical soliton pulse propagation in a twin-core nonlinear optical fiber. Through the Kadomtsev-Petviashvili hierarchy reduction, we present the bright-dark and dark-dark soliton solutions in terms of the Grammian for such equations. With the help of analytic and graphic analysis, head-on and overtaking elastic interactions between the two solitons are presented, as well as the bound-state solitons. Particularly, we find the inelastic interaction between the bright-dark two solitons. One of the electromagnetic fields presents the V-shape profile, while the other one presents the Y-shape profile.

Nonlinear recurrent neural networks for finite-time solution of general time-varying linear matrix equations.

PubMed

Xiao, Lin; Liao, Bolin; Li, Shuai; Chen, Ke

2018-02-01

In order to solve general time-varying linear matrix equations (LMEs) more efficiently, this paper proposes two nonlinear recurrent neural networks based on two nonlinear activation functions. According to Lyapunov theory, such two nonlinear recurrent neural networks are proved to be convergent within finite-time. Besides, by solving differential equation, the upper bounds of the finite convergence time are determined analytically. Compared with existing recurrent neural networks, the proposed two nonlinear recurrent neural networks have a better convergence property (i.e., the upper bound is lower), and thus the accurate solutions of general time-varying LMEs can be obtained with less time. At last, various different situations have been considered by setting different coefficient matrices of general time-varying LMEs and a great variety of computer simulations (including the application to robot manipulators) have been conducted to validate the better finite-time convergence of the proposed two nonlinear recurrent neural networks. Copyright © 2017 Elsevier Ltd. All rights reserved.

Multiple solutions to a magnetic nonlinear Choquard equation

NASA Astrophysics Data System (ADS)

Cingolani, Silvia; Clapp, Mónica; Secchi, Simone

2012-04-01

We consider the stationary nonlinear magnetic Choquard equation (- inabla+ A(x))2u + V (x)u = (1/|x|^{α}ast |u|pright) |u|^{p-2}u,quad xin{R}N where A is a real-valued vector potential, V is a real-valued scalar potential, N ≥ 3, {α in (0, N)} and 2 - ( α/ N) < p < (2 N - α)/( N-2). We assume that both A and V are compatible with the action of some group G of linear isometries of {{R}N} . We establish the existence of multiple complex valued solutions to this equation which satisfy the symmetry condition u(gx) = tau(g)u(x)quad{for all } g in G,x in {R}N, where {tau : G rightarrow {S}1} is a given group homomorphism into the unit complex numbers.

A general formula for Rayleigh-Schroedinger perturbation energy utilizing a power series expansion of the quantum mechanical Hamiltonian

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

Herbert, J.M.

1997-02-01

Perturbation theory has long been utilized by quantum chemists as a method for approximating solutions to the Schroedinger equation. Perturbation treatments represent a system`s energy as a power series in which each additional term further corrects the total energy; it is therefore convenient to have an explicit formula for the nth-order energy correction term. If all perturbations are collected into a single Hamiltonian operator, such a closed-form expression for the nth-order energy correction is well known; however, use of a single perturbed Hamiltonian often leads to divergent energy series, while superior convergence behavior is obtained by expanding the perturbed Hamiltonianmore » in a power series. This report presents a closed-form expression for the nth-order energy correction obtained using Rayleigh-Schroedinger perturbation theory and a power series expansion of the Hamiltonian.« less

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

NASA Astrophysics Data System (ADS)

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

2017-11-01

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

Formation of quasiparallel Alfven solitons

NASA Technical Reports Server (NTRS)

Hamilton, R. L.; Kennel, C. F.; Mjolhus, E.

1992-01-01

The formation of quasi-parallel Alfven solitons is investigated through the inverse scattering transformation (IST) for the derivative nonlinear Schroedinger (DNLS) equation. The DNLS has a rich complement of soliton solutions consisting of a two-parameter soliton family and a one-parameter bright/dark soliton family. In this paper, the physical roles and origins of these soliton families are inferred through an analytic study of the scattering data generated by the IST for a set of initial profiles. The DNLS equation has as limiting forms the nonlinear Schroedinger (NLS), Korteweg-de-Vries (KdV) and modified Korteweg-de-Vries (MKdV) equations. Each of these limits is briefly reviewed in the physical context of quasi-parallel Alfven waves. The existence of these limiting forms serves as a natural framework for discussing the formation of Alfven solitons.

A Bohmian approach to the non-Markovian non-linear Schrödinger–Langevin equation

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

Vargas, Andrés F.; Morales-Durán, Nicolás; Bargueño, Pedro, E-mail: p.bargueno@uniandes.edu.co

2015-05-15

In this work, a non-Markovian non-linear Schrödinger–Langevin equation is derived from the system-plus-bath approach. After analyzing in detail previous Markovian cases, Bohmian mechanics is shown to be a powerful tool for obtaining the desired generalized equation.

Solving Nonlinear Fractional Differential Equation by Generalized Mittag-Leffler Function Method

NASA Astrophysics Data System (ADS)

Arafa, A. A. M.; Rida, S. Z.; Mohammadein, A. A.; Ali, H. M.

2013-06-01

In this paper, we use Mittag—Leffler function method for solving some nonlinear fractional differential equations. A new solution is constructed in power series. The fractional derivatives are described by Caputo's sense. To illustrate the reliability of the method, some examples are provided.

Bound vector solitons and soliton complexes for the coupled nonlinear Schrödinger equations.

PubMed

Sun, Zhi-Yuan; Gao, Yi-Tian; Yu, Xin; Liu, Wen-Jun; Liu, Ying

2009-12-01

Dynamic features describing the collisions of the bound vector solitons and soliton complexes are investigated for the coupled nonlinear Schrödinger (CNLS) equations, which model the propagation of the multimode soliton pulses under some physical situations in nonlinear fiber optics. Equations of such type have also been seen in water waves and plasmas. By the appropriate choices of the arbitrary parameters for the multisoliton solutions derived through the Hirota bilinear method, the periodic structures along the propagation are classified according to the relative relations of the real wave numbers. Furthermore, parameters are shown to control the intensity distributions and interaction patterns for the bound vector solitons and soliton complexes. Transformations of the soliton types (shape changing with intensity redistribution) during the collisions of those stationary structures with the regular one soliton are discussed, in which a class of inelastic properties is involved. Discussions could be expected to be helpful in interpreting such structures in the multimode nonlinear fiber optics and equally applied to other systems governed by the CNLS equations, e.g., the plasma physics and Bose-Einstein condensates.

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

PubMed

Li, Li; Yu, Fajun

2017-09-06

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

An efficient nonlinear finite-difference approach in the computational modeling of the dynamics of a nonlinear diffusion-reaction equation in microbial ecology.

PubMed

Macías-Díaz, J E; Macías, Siegfried; Medina-Ramírez, I E

2013-12-01

In this manuscript, we present a computational model to approximate the solutions of a partial differential equation which describes the growth dynamics of microbial films. The numerical technique reported in this work is an explicit, nonlinear finite-difference methodology which is computationally implemented using Newton's method. Our scheme is compared numerically against an implicit, linear finite-difference discretization of the same partial differential equation, whose computer coding requires an implementation of the stabilized bi-conjugate gradient method. Our numerical results evince that the nonlinear approach results in a more efficient approximation to the solutions of the biofilm model considered, and demands less computer memory. Moreover, the positivity of initial profiles is preserved in the practice by the nonlinear scheme proposed. Copyright © 2013 Elsevier Ltd. All rights reserved.

Exact axisymmetric solutions of the Maxwell equations in a nonlinear nondispersive medium.

PubMed

Petrov, E Yu; Kudrin, A V

2010-05-14

The features of propagation of intense waves are of great interest for theory and experiment in electrodynamics and acoustics. The behavior of nonlinear waves in a bounded volume is of special importance and, at the same time, is an extremely complicated problem. It seems almost impossible to find a rigorous solution to such a problem even for any model of nonlinearity. We obtain the first exact solution of this type. We present a new method for deriving exact solutions of the Maxwell equations in a nonlinear medium without dispersion and give examples of the obtained solutions that describe propagation of cylindrical electromagnetic waves in a nonlinear nondispersive medium and free electromagnetic oscillations in a cylindrical cavity resonator filled with such a medium.

Application of variational and Galerkin equations to linear and nonlinear finite element analysis

NASA Technical Reports Server (NTRS)

Yu, Y.-Y.

1974-01-01

The paper discusses the application of the variational equation to nonlinear finite element analysis. The problem of beam vibration with large deflection is considered. The variational equation is shown to be flexible in both the solution of a general problem and in the finite element formulation. Difficulties are shown to arise when Galerkin's equations are used in the consideration of the finite element formulation of two-dimensional linear elasticity and of the linear classical beam.

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.

A note on the generation of phase plane plots on a digital computer. [for solution of nonlinear differential equations

NASA Technical Reports Server (NTRS)

Simon, M. K.

1980-01-01

A technique is presented for generating phase plane plots on a digital computer which circumvents the difficulties associated with more traditional methods of numerical solving nonlinear differential equations. In particular, the nonlinear differential equation of operation is formulated.

Nonlinear flap-lag-axial equations of a rotating beam with arbitrary precone angle

NASA Technical Reports Server (NTRS)

Kvaternik, R. G.; White, W. F., Jr.; Kaza, K. R. V.

1978-01-01

In an attempt both to unify and extend the analytical basis of several aspects of the dynamic behavior of flexible rotating beams, the second-degree nonlinear equations of motion for the coupled flapwise bending, lagwise bending, and axial extension of an untwisted, torsionally rigid, nonuniform, rotating beam having an arbitrary angle of precone with the plane perpendicular to the axis of rotation are derived using Hamilton's principle. The derivation of the equations is based on the geometric nonlinear theory of elasticity and the resulting equations are consistent with the assumption that the strains are negligible compared to unity. No restrictions are imposed on the relative displacements or angular rotations of the cross sections of the beam other than those implied by the assumption of small strains. Illustrative numerical results, obtained by using an integrating matrix as the basis for the method of solution, are presented both for the purpose of validating the present method of solution and indicating the range of applicability of the equations of motion and the method of solution.

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

NASA Astrophysics Data System (ADS)

Yan, Zhenya; Bluman, George

2002-11-01

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

3D early embryogenesis image filtering by nonlinear partial differential equations.

PubMed

Krivá, Z; Mikula, K; Peyriéras, N; Rizzi, B; Sarti, A; Stasová, O

2010-08-01

We present nonlinear diffusion equations, numerical schemes to solve them and their application for filtering 3D images obtained from laser scanning microscopy (LSM) of living zebrafish embryos, with a goal to identify the optimal filtering method and its parameters. In the large scale applications dealing with analysis of 3D+time embryogenesis images, an important objective is a correct detection of the number and position of cell nuclei yielding the spatio-temporal cell lineage tree of embryogenesis. The filtering is the first and necessary step of the image analysis chain and must lead to correct results, removing the noise, sharpening the nuclei edges and correcting the acquisition errors related to spuriously connected subregions. In this paper we study such properties for the regularized Perona-Malik model and for the generalized mean curvature flow equations in the level-set formulation. A comparison with other nonlinear diffusion filters, like tensor anisotropic diffusion and Beltrami flow, is also included. All numerical schemes are based on the same discretization principles, i.e. finite volume method in space and semi-implicit scheme in time, for solving nonlinear partial differential equations. These numerical schemes are unconditionally stable, fast and naturally parallelizable. The filtering results are evaluated and compared first using the Mean Hausdorff distance between a gold standard and different isosurfaces of original and filtered data. Then, the number of isosurface connected components in a region of interest (ROI) detected in original and after the filtering is compared with the corresponding correct number of nuclei in the gold standard. Such analysis proves the robustness and reliability of the edge preserving nonlinear diffusion filtering for this type of data and lead to finding the optimal filtering parameters for the studied models and numerical schemes. Further comparisons consist in ability of splitting the very close objects which

Quasi-periodic solutions of a quasi-periodically forced nonlinear beam equation

NASA Astrophysics Data System (ADS)

Wang, Yi

2012-06-01

In this paper, one quasi-periodically forced nonlinear beam equation utt+uxxxx+μu+ɛg(ωt,x)u3=0,μ>0,x∈[0,π] with hinged boundary conditions is considered. Here ɛ is a small positive parameter, g( ωt, x) is real analytic in all variables and quasi-periodic in t with a frequency vector ω = ( ω1, ω2, … , ωm). It is proved that the above equation admits small-amplitude quasi-periodic solutions.

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

PubMed

Kanagawa, Tetsuya

2015-05-01

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

Reduced-order modelling of parameter-dependent, linear and nonlinear dynamic partial differential equation models

PubMed Central

Xing, W. W.; Triantafyllidis, V.

2017-01-01

In this paper, we develop reduced-order models for dynamic, parameter-dependent, linear and nonlinear partial differential equations using proper orthogonal decomposition (POD). The main challenges are to accurately and efficiently approximate the POD bases for new parameter values and, in the case of nonlinear problems, to efficiently handle the nonlinear terms. We use a Bayesian nonlinear regression approach to learn the snapshots of the solutions and the nonlinearities for new parameter values. Computational efficiency is ensured by using manifold learning to perform the emulation in a low-dimensional space. The accuracy of the method is demonstrated on a linear and a nonlinear example, with comparisons with a global basis approach. PMID:28484327

Quantum spatial propagation of squeezed light in a degenerate parametric amplifier

NASA Technical Reports Server (NTRS)

Deutsch, Ivan H.; Garrison, John C.

1992-01-01

Differential equations which describe the steady state spatial evolution of nonclassical light are established using standard quantum field theoretic techniques. A Schroedinger equation for the state vector of the optical field is derived using the quantum analog of the slowly varying envelope approximation (SVEA). The steady state solutions are those that satisfy the time independent Schroedinger equation. The resulting eigenvalue problem then leads to the spatial propagation equations. For the degenerate parametric amplifier this method shows that the squeezing parameter obey nonlinear differential equations coupled by the amplifier gain and phase mismatch. The solution to these differential equations is equivalent to one obtained from the classical three wave mixing steady state solution to the parametric amplifier with a nondepleted pump.

On new classes of solutions of nonlinear partial differential equations in the form of convergent special series

NASA Astrophysics Data System (ADS)

Filimonov, M. Yu.

2017-12-01

The method of special series with recursively calculated coefficients is used to solve nonlinear partial differential equations. The recurrence of finding the coefficients of the series is achieved due to a special choice of functions, in powers of which the solution is expanded in a series. We obtain a sequence of linear partial differential equations to find the coefficients of the series constructed. In many cases, one can deal with a sequence of linear ordinary differential equations. We construct classes of solutions in the form of convergent series for a certain class of nonlinear evolution equations. A new class of solutions of generalized Boussinesque equation with an arbitrary function in the form of a convergent series is constructed.

A novel technique to solve nonlinear higher-index Hessenberg differential-algebraic equations by Adomian decomposition method.

PubMed

Benhammouda, Brahim

2016-01-01

Since 1980, the Adomian decomposition method (ADM) has been extensively used as a simple powerful tool that applies directly to solve different kinds of nonlinear equations including functional, differential, integro-differential and algebraic equations. However, for differential-algebraic equations (DAEs) the ADM is applied only in four earlier works. There, the DAEs are first pre-processed by some transformations like index reductions before applying the ADM. The drawback of such transformations is that they can involve complex algorithms, can be computationally expensive and may lead to non-physical solutions. The purpose of this paper is to propose a novel technique that applies the ADM directly to solve a class of nonlinear higher-index Hessenberg DAEs systems efficiently. The main advantage of this technique is that; firstly it avoids complex transformations like index reductions and leads to a simple general algorithm. Secondly, it reduces the computational work by solving only linear algebraic systems with a constant coefficient matrix at each iteration, except for the first iteration where the algebraic system is nonlinear (if the DAE is nonlinear with respect to the algebraic variable). To demonstrate the effectiveness of the proposed technique, we apply it to a nonlinear index-three Hessenberg DAEs system with nonlinear algebraic constraints. This technique is straightforward and can be programmed in Maple or Mathematica to simulate real application problems.

Symmetry Reductions of Fourth-Order Nonlinear Diffusion Equations: Lubrication Model and Some Generalizations

NASA Astrophysics Data System (ADS)

Gandarias, M. L.; Medina, E.

Fourth-order nonlinear diffusion equations appear frequently in the description of physical processes, among these, the lubrication equation ut = (unuxxxx)x or the corresponding modified version ut = unuxxxx play an important role in the study of the interface movements. In this work we analyze the generalizations of the above equations given by ut = (f(u)uxxxx)x, ut = (f(u)uxxxx, and we find that if f(u) = un or f(u) = e-u the equations admit extra classical symmetries. The corresponding reductions are performed and some solutions are characterized.

Detecting Moving Targets by Use of Soliton Resonances

NASA Technical Reports Server (NTRS)

Zak, Michael; Kulikov, Igor

2003-01-01

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

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

NASA Astrophysics Data System (ADS)

Seadawy, Aly R.

2017-01-01

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

Nonlinear Interaction of Detuned Instability Waves in Boundary-Layer Transition: Amplitude Equations

NASA Technical Reports Server (NTRS)

Lee, Sang Soo

1998-01-01

The non-equilibrium critical-layer analysis of a system of frequency-detuned resonant-triads is presented. In this part of the analysis, the system of partial differential critical-layer equations derived in Part I is solved analytically to yield the amplitude equations which are analyzed using a combination of asymptotic and numerical methods. Numerical solutions of the inviscid non-equilibrium oblique-mode amplitude equations show that the frequency-detuned self-interaction enhances the growth of the lower-frequency oblique modes more than the higher-frequency ones. All amplitudes become singular at the same finite downstream position. The frequency detuning delays the occurrence of the singularity. The spanwise-periodic mean-flow distortion and low-frequency nonlinear modes are generated by the critical-layer interaction between frequency-detuned oblique modes. The nonlinear mean flow and higher harmonics as well as the primary instabilities become as large as the base mean flow in the inviscid wall layer in the downstream region where the distance from the singularity is of the order of the wavelength scale.

Nonlinear Riccati equations as a unifying link between linear quantum mechanics and other fields of physics

NASA Astrophysics Data System (ADS)

Schuch, Dieter

2014-04-01

Theoretical physics seems to be in a kind of schizophrenic state. Many phenomena in the observable macroscopic world obey nonlinear evolution equations, whereas 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. I claim that linearity in quantum mechanics is not as essential as it apparently seems since quantum mechanics can be reformulated in terms of nonlinear Riccati equations. In a first step, it will be shown where complex Riccati equations appear in time-dependent quantum mechanics and how they can be treated and compared with similar space-dependent Riccati equations in supersymmetric quantum mechanics. Furthermore, the time-independent Schrödinger equation can also be rewritten as a complex Riccati equation. Finally, it will be shown that (real and complex) Riccati equations also appear in many other fields of physics, like statistical thermodynamics and cosmology.

Collapse for the higher-order nonlinear Schrödinger equation

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

Achilleos, V.; Diamantidis, S.; Frantzeskakis, D. J.

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

Collapse for the higher-order nonlinear Schrödinger equation

DOE PAGES

Achilleos, V.; Diamantidis, S.; Frantzeskakis, D. J.; ...

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

Generalized nonlinear Schrödinger equation and ultraslow optical solitons in a cold four-state atomic system.

PubMed

Hang, Chao; Huang, Guoxiang; Deng, L

2006-03-01

We investigate the influence of high-order dispersion and nonlinearity on the propagation of ultraslow optical solitons in a lifetime broadened four-state atomic system under a Raman excitation. Using a standard method of multiple-scales we derive a generalized nonlinear Schrödinger equation and show that for realistic physical parameters and at the pulse duration of 10(-6)s, the effects of third-order linear dispersion, nonlinear dispersion, and delay in nonlinear refractive index can be significant and may not be considered as perturbations. We provide exact soliton solutions for the generalized nonlinear Schrödinger equation and demonstrate that optical solitons obtained may still have ultraslow propagating velocity. Numerical simulations on the stability and interaction of these ultraslow optical solitons in the presence of linear and differential absorptions are also presented.

Pair correlation function and nonlinear kinetic equation for a spatially uniform polarizable nonideal plasma

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

Belyi, V.V.; Kukharenko, Y.A.; Wallenborn, J.

Taking into account the first non-Markovian correction to the Balescu-Lenard equation, we have derived an expression for the pair correlation function and a nonlinear kinetic equation valid for a nonideal polarized classical plasma. This last equation allows for the description of the correlational energy evolution and shows the global conservation of energy with dynamical polarization. {copyright} {ital 1996 The American Physical Society.}

Optical solitons and modulation instability analysis with (3 + 1)-dimensional nonlinear Shrödinger equation

NASA Astrophysics Data System (ADS)

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

2017-12-01

This paper addresses the (3 + 1)-dimensional nonlinear Shrödinger equation (NLSE) that serves as the model to study the propagation of optical solitons through nonlinear optical fibers. Two integration schemes are employed to study the equation. These are the complex envelope function ansatz and the solitary wave ansatz with Jaccobi elliptic function methods, we present the exact dark, bright and dark-bright or combined optical solitons to the model. The intensity as well as the nonlinear phase shift of the solitons are reported. The modulation instability aspects are discussed using the concept of linear stability analysis. The MI gain is got. Numerical simulation of the obtained results are analyzed with interesting figures showing the physical meaning of the solutions.

The nonlinear wave equation for higher harmonics in free-electron lasers

NASA Technical Reports Server (NTRS)

Colson, W. B.

1981-01-01

The nonlinear wave equation and self-consistent pendulum equation are generalized to describe free-electron laser operation in higher harmonics; this can significantly extend their tunable range to shorter wavelengths. The dynamics of the laser field's amplitude and phase are explored for a wide range of parameters using families of normalized gain curves applicable to both the fundamental and harmonics. The electron phase-space displays the fundamental physics driving the wave, and this picture is used to distinguish between the effects of high gain and Coulomb forces.

Generalized Lagrange Jacobi Gauss-Lobatto (GLJGL) Collocation Method for Solving Linear and Nonlinear Fokker-Planck Equations

NASA Astrophysics Data System (ADS)

Parand, K.; Latifi, S.; Moayeri, M. M.; Delkhosh, M.

2018-05-01

In this study, we have constructed a new numerical approach for solving the time-dependent linear and nonlinear Fokker-Planck equations. In fact, we have discretized the time variable with Crank-Nicolson method and for the space variable, a numerical method based on Generalized Lagrange Jacobi Gauss-Lobatto (GLJGL) collocation method is applied. It leads to in solving the equation in a series of time steps and at each time step, the problem is reduced to a problem consisting of a system of algebraic equations that greatly simplifies the problem. One can observe that the proposed method is simple and accurate. Indeed, one of its merits is that it is derivative-free and by proposing a formula for derivative matrices, the difficulty aroused in calculation is overcome, along with that it does not need to calculate the General Lagrange basis and matrices; they have Kronecker property. Linear and nonlinear Fokker-Planck equations are given as examples and the results amply demonstrate that the presented method is very valid, effective, reliable and does not require any restrictive assumptions for nonlinear terms.

Numerical solution of the general coupled nonlinear Schrödinger equations on unbounded domains.

PubMed

Li, Hongwei; Guo, Yue

2017-12-01

The numerical solution of the general coupled nonlinear Schrödinger equations on unbounded domains is considered by applying the artificial boundary method in this paper. In order to design the local absorbing boundary conditions for the coupled nonlinear Schrödinger equations, we generalize the unified approach previously proposed [J. Zhang et al., Phys. Rev. E 78, 026709 (2008)PLEEE81539-375510.1103/PhysRevE.78.026709]. Based on the methodology underlying the unified approach, the original problem is split into two parts, linear and nonlinear terms, and we then achieve a one-way operator to approximate the linear term to make the wave out-going, and finally we combine the one-way operator with the nonlinear term to derive the local absorbing boundary conditions. Then we reduce the original problem into an initial boundary value problem on the bounded domain, which can be solved by the finite difference method. The stability of the reduced problem is also analyzed by introducing some auxiliary variables. Ample numerical examples are presented to verify the accuracy and effectiveness of our proposed method.

Bayesian Analysis of Nonlinear Structural Equation Models with Nonignorable Missing Data

ERIC Educational Resources Information Center

Lee, Sik-Yum

2006-01-01

A Bayesian approach is developed for analyzing nonlinear structural equation models with nonignorable missing data. The nonignorable missingness mechanism is specified by a logistic regression model. A hybrid algorithm that combines the Gibbs sampler and the Metropolis-Hastings algorithm is used to produce the joint Bayesian estimates of…

Numerical solution of nonlinear partial differential equations of mixed type. [finite difference approximation

NASA Technical Reports Server (NTRS)

Jameson, A.

1976-01-01

A review is presented of some recently developed numerical methods for the solution of nonlinear equations of mixed type. The methods considered use finite difference approximations to the differential equation. Central difference formulas are employed in the subsonic zone and upwind difference formulas are used in the supersonic zone. The relaxation method for the small disturbance equation is discussed and a description is given of difference schemes for the potential flow equation in quasi-linear form. Attention is also given to difference schemes for the potential flow equation in conservation form, the analysis of relaxation schemes by the time dependent analogy, the accelerated iterative method, and three-dimensional calculations.

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

Optical rogue waves for the inhomogeneous generalized nonlinear Schrödinger equation.

PubMed

Loomba, Shally; Kaur, Harleen

2013-12-01

We present optical rogue wave solutions for a generalized nonlinear Schrodinger equation by using similarity transformation. We have predicted the propagation of rogue waves through a nonlinear optical fiber for three cases: (i) dispersion increasing (decreasing) fiber, (ii) periodic dispersion parameter, and (iii) hyperbolic dispersion parameter. We found that the rogue waves and their interactions can be tuned by properly choosing the parameters. We expect that our results can be used to realize improved signal transmission through optical rogue waves.

Periodic solutions of second-order nonlinear difference equations containing a small parameter. III - Perturbation theory

NASA Technical Reports Server (NTRS)

Mickens, R. E.

1986-01-01

A technique to construct a uniformly valid perturbation series solution to a particular class of nonlinear difference equations is shown. The method allows the determination of approximations to the periodic solutions to these equations. An example illustrating the technique is presented.

Liouville type theorems of a nonlinear elliptic equation for the V-Laplacian

NASA Astrophysics Data System (ADS)

Huang, Guangyue; Li, Zhi

2018-03-01

In this paper, we consider Liouville type theorems for positive solutions to the following nonlinear elliptic equation: Δ _V u+aulog u=0, where a is a nonzero real constant. By using gradient estimates, we obtain upper bounds of |\

Alpha models for rotating Navier-Stokes equations in geophysics with nonlinear dispersive regularization

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.

High-order rogue waves in vector nonlinear Schrödinger equations.

PubMed

Ling, Liming; Guo, Boling; Zhao, Li-Chen

2014-04-01

We study the dynamics of high-order rogue waves (RWs) in two-component coupled nonlinear Schrödinger equations. We find that four fundamental rogue waves can emerge from second-order vector RWs in the coupled system, in contrast to the high-order ones in single-component systems. The distribution shape can be quadrilateral, triangle, and line structures by varying the proper initial excitations given by the exact analytical solutions. The distribution pattern for vector RWs is more abundant than that for scalar rogue waves. Possibilities to observe these new patterns for rogue waves are discussed for a nonlinear fiber.

Piecewise linear emulator of the nonlinear Schrödinger equation and the resulting analytic solutions for Bose-Einstein condensates.

PubMed

Theodorakis, Stavros

2003-06-01

We emulate the cubic term Psi(3) in the nonlinear Schrödinger equation by a piecewise linear term, thus reducing the problem to a set of uncoupled linear inhomogeneous differential equations. The resulting analytic expressions constitute an excellent approximation to the exact solutions, as is explicitly shown in the case of the kink, the vortex, and a delta function trap. Such a piecewise linear emulation can be used for any differential equation where the only nonlinearity is a Psi(3) one. In particular, it can be used for the nonlinear Schrödinger equation in the presence of harmonic traps, giving analytic Bose-Einstein condensate solutions that reproduce very accurately the numerically calculated ones in one, two, and three dimensions.

Moving solitons in the discrete nonlinear Schrödinger equation.

PubMed

Oxtoby, O F; Barashenkov, I V

2007-09-01

Using the method of asymptotics beyond all orders, we evaluate the amplitude of radiation from a moving small-amplitude soliton in the discrete nonlinear Schrödinger equation. When the nonlinearity is of the cubic type, this amplitude is shown to be nonzero for all velocities and therefore small-amplitude solitons moving without emitting radiation do not exist. In the case of a saturable nonlinearity, on the other hand, the radiation is found to be completely suppressed when the soliton moves at one of certain isolated "sliding velocities." We show that a discrete soliton moving at a general speed will experience radiative deceleration until it either stops and remains pinned to the lattice or--in the saturable case--locks, metastably, onto one of the sliding velocities. When the soliton's amplitude is small, however, this deceleration is extremely slow; hence, despite losing energy to radiation, the discrete soliton may spend an exponentially long time traveling with virtually unchanged amplitude and speed.

Orbital stability and energy estimate of ground states of saturable nonlinear Schrödinger equations with intensity functions in R2

NASA Astrophysics Data System (ADS)

Lin, Tai-Chia; Wang, Xiaoming; Wang, Zhi-Qiang

2017-10-01

Conventionally, the existence and orbital stability of ground states of nonlinear Schrödinger (NLS) equations with power-law nonlinearity (subcritical case) can be proved by an argument using strict subadditivity of the ground state energy and the concentration compactness method of Cazenave and Lions [4]. However, for saturable nonlinearity, such an argument is not applicable because strict subadditivity of the ground state energy fails in this case. Here we use a convexity argument to prove the existence and orbital stability of ground states of NLS equations with saturable nonlinearity and intensity functions in R2. Besides, we derive the energy estimate of ground states of saturable NLS equations with intensity functions using the eigenvalue estimate of saturable NLS equations without intensity function.

Nonlinear integral equations for the sausage model

NASA Astrophysics Data System (ADS)

Ahn, Changrim; Balog, Janos; Ravanini, Francesco

2017-08-01

The sausage model, first proposed by Fateev, Onofri, and Zamolodchikov, is a deformation of the O(3) sigma model preserving integrability. The target space is deformed from the sphere to ‘sausage’ shape by a deformation parameter ν. This model is defined by a factorizable S-matrix which is obtained by deforming that of the O(3) sigma model by a parameter λ. Clues for the deformed sigma model are provided by various UV and IR information through the thermodynamic Bethe ansatz (TBA) analysis based on the S-matrix. Application of TBA to the sausage model is, however, limited to the case of 1/λ integer where the coupled integral equations can be truncated to a finite number. In this paper, we propose a finite set of nonlinear integral equations (NLIEs), which are applicable to generic value of λ. Our derivation is based on T-Q relations extracted from the truncated TBA equations. For a consistency check, we compute next-leading order corrections of the vacuum energy and extract the S-matrix information in the IR limit. We also solved the NLIE both analytically and numerically in the UV limit to get the effective central charge and compared with that of the zero-mode dynamics to obtain exact relation between ν and λ. Dedicated to the memory of Petr Petrovich Kulish.

Solution of the nonlinear mixed Volterra-Fredholm integral equations by hybrid of block-pulse functions and Bernoulli polynomials.

PubMed

Mashayekhi, S; Razzaghi, M; Tripak, O

2014-01-01

A new numerical method for solving the nonlinear mixed Volterra-Fredholm integral equations is presented. This method is based upon hybrid functions approximation. The properties of hybrid functions consisting of block-pulse functions and Bernoulli polynomials are presented. The operational matrices of integration and product are given. These matrices are then utilized to reduce the nonlinear mixed Volterra-Fredholm integral equations to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

Solution of the Nonlinear Mixed Volterra-Fredholm Integral Equations by Hybrid of Block-Pulse Functions and Bernoulli Polynomials

PubMed Central

Mashayekhi, S.; Razzaghi, M.; Tripak, O.

2014-01-01

A new numerical method for solving the nonlinear mixed Volterra-Fredholm integral equations is presented. This method is based upon hybrid functions approximation. The properties of hybrid functions consisting of block-pulse functions and Bernoulli polynomials are presented. The operational matrices of integration and product are given. These matrices are then utilized to reduce the nonlinear mixed Volterra-Fredholm integral equations to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique. PMID:24523638

Solution of a few nonlinear problems in aerodynamics by the finite elements and functional least squares methods. Ph.D. Thesis - Paris Univ.; [mathematical models of transonic flow using nonlinear equations

NASA Technical Reports Server (NTRS)

Periaux, J.

1979-01-01

The numerical simulation of the transonic flows of idealized fluids and of incompressible viscous fluids, by the nonlinear least squares methods is presented. The nonlinear equations, the boundary conditions, and the various constraints controlling the two types of flow are described. The standard iterative methods for solving a quasi elliptical nonlinear equation with partial derivatives are reviewed with emphasis placed on two examples: the fixed point method applied to the Gelder functional in the case of compressible subsonic flows and the Newton method used in the technique of decomposition of the lifting potential. The new abstract least squares method is discussed. It consists of substituting the nonlinear equation by a problem of minimization in a H to the minus 1 type Sobolev functional space.

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.

Action principle for overdetermined systems of nonlinear field equations

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

Nissimov, E.; Pacheva, S.; Solomon, S.

1989-01-01

The authors propose a general scheme for constructing an action principle for arbitrary consistent overdetermined systems of nonlinear field equations. The principal tool is the BFV-BRST formalism. There is no need for star-product nor Chern-Simons forms. The main application of this general construction is the derivation of a superspace action in terms of unconstrained superfields for the D = 10N = 1 Super-Yang-Mills theory. The latter contains cubic as well as quartic interactions.

Renormalized vibrations and normal energy transport in 1d FPU-like discrete nonlinear Schrödinger equations.

PubMed

Li, Simeng; Li, Nianbei

2018-03-28

For one-dimensional (1d) nonlinear atomic lattices, the models with on-site nonlinearities such as the Frenkel-Kontorova (FK) and ϕ 4 lattices have normal energy transport while the models with inter-site nonlinearities such as the Fermi-Pasta-Ulam-β (FPU-β) lattice exhibit anomalous energy transport. The 1d Discrete Nonlinear Schrödinger (DNLS) equations with on-site nonlinearities has been previously studied and normal energy transport has also been found. Here, we investigate the energy transport of 1d FPU-like DNLS equations with inter-site nonlinearities. Extended from the FPU-β lattice, the renormalized vibration theory is developed for the FPU-like DNLS models and the predicted renormalized vibrations are verified by direct numerical simulations same as the FPU-β lattice. However, the energy diffusion processes are explored and normal energy transport is observed for the 1d FPU-like DNLS models, which is different from their atomic lattice counterpart of FPU-β lattice. The reason might be that, unlike nonlinear atomic lattices where models with on-site nonlinearities have one less conserved quantities than the models with inter-site nonlinearities, the DNLS models with on-site or inter-site nonlinearities have the same number of conserved quantities as the result of gauge transformation.

Unleashing Empirical Equations with "Nonlinear Fitting" and "GUM Tree Calculator"

NASA Astrophysics Data System (ADS)

Lovell-Smith, J. W.; Saunders, P.; Feistel, R.

2017-10-01

Empirical equations having large numbers of fitted parameters, such as the international standard reference equations published by the International Association for the Properties of Water and Steam (IAPWS), which form the basis of the "Thermodynamic Equation of Seawater—2010" (TEOS-10), provide the means to calculate many quantities very accurately. The parameters of these equations are found by least-squares fitting to large bodies of measurement data. However, the usefulness of these equations is limited since uncertainties are not readily available for most of the quantities able to be calculated, the covariance of the measurement data is not considered, and further propagation of the uncertainty in the calculated result is restricted since the covariance of calculated quantities is unknown. In this paper, we present two tools developed at MSL that are particularly useful in unleashing the full power of such empirical equations. "Nonlinear Fitting" enables propagation of the covariance of the measurement data into the parameters using generalized least-squares methods. The parameter covariance then may be published along with the equations. Then, when using these large, complex equations, "GUM Tree Calculator" enables the simultaneous calculation of any derived quantity and its uncertainty, by automatic propagation of the parameter covariance into the calculated quantity. We demonstrate these tools in exploratory work to determine and propagate uncertainties associated with the IAPWS-95 parameters.

Continuations of the nonlinear Schrödinger equation beyond the singularity

NASA Astrophysics Data System (ADS)

Fibich, G.; Klein, M.

2011-07-01

We present four continuations of the critical nonlinear Schrödinger equation (NLS) beyond the singularity: (1) a sub-threshold power continuation, (2) a shrinking-hole continuation for ring-type solutions, (3) a vanishing nonlinear-damping continuation and (4) a complex Ginzburg-Landau (CGL) continuation. Using asymptotic analysis, we explicitly calculate the limiting solutions beyond the singularity. These calculations show that for generic initial data that lead to a loglog collapse, the sub-threshold power limit is a Bourgain-Wang solution, both before and after the singularity, and the vanishing nonlinear-damping and CGL limits are a loglog solution before the singularity, and have an infinite-velocity expanding core after the singularity. Our results suggest that all NLS continuations share the universal feature that after the singularity time Tc, the phase of the singular core is only determined up to multiplication by eiθ. As a result, interactions between post-collapse beams (filaments) become chaotic. We also show that when the continuation model leads to a point singularity and preserves the NLS invariance under the transformation t → -t and ψ → ψ*, the singular core of the weak solution is symmetric with respect to Tc. Therefore, the sub-threshold power and the shrinking-hole continuations are symmetric with respect to Tc, but continuations which are based on perturbations of the NLS equation are generically asymmetric.

Exact Solutions to Several Nonlinear Cases of Generalized Grad-Shafranov Equation for Ideal Magnetohydrodynamic Flows in Axisymmetric Domain

NASA Astrophysics Data System (ADS)

Adem, Abdullahi Rashid; Moawad, Salah M.

2018-05-01

In this paper, the steady-state equations of ideal magnetohydrodynamic incompressible flows in axisymmetric domains are investigated. These flows are governed by a second-order elliptic partial differential equation as a type of generalized Grad-Shafranov equation. The problem of finding exact equilibria to the full governing equations in the presence of incompressible mass flows is considered. Two different types of constraints on position variables are presented to construct exact solution classes for several nonlinear cases of the governing equations. Some of the obtained results are checked for their applications to magnetic confinement plasma. Besides, they cover many previous configurations and include new considerations about the nonlinearity of magnetic flux stream variables.

A Family of Ellipse Methods for Solving Non-Linear Equations

ERIC Educational Resources Information Center

Gupta, K. C.; Kanwar, V.; Kumar, Sanjeev

2009-01-01

This note presents a method for the numerical approximation of simple zeros of a non-linear equation in one variable. In order to do so, the method uses an ellipse rather than a tangent approach. The main advantage of our method is that it does not fail even if the derivative of the function is either zero or very small in the vicinity of the…

On a new class of completely integrable nonlinear wave equations. I. Infinitely many conservation laws

NASA Astrophysics Data System (ADS)

Nutku, Y.

1985-06-01

We point out a class of nonlinear wave equations which admit infinitely many conserved quantities. These equations are characterized by a pair of exact one-forms. The implication that they are closed gives rise to equations, the characteristics and Riemann invariants of which are readily obtained. The construction of the conservation laws requires the solution of a linear second-order equation which can be reduced to canonical form using the Riemann invariants. The hodograph transformation results in a similar linear equation. We discuss also the symplectic structure and Bäcklund transformations associated with these equations.

Exact and Approximate Statistical Inference for Nonlinear Regression and the Estimating Equation Approach.

PubMed

Demidenko, Eugene

2017-09-01

The exact density distribution of the nonlinear least squares estimator in the one-parameter regression model is derived in closed form and expressed through the cumulative distribution function of the standard normal variable. Several proposals to generalize this result are discussed. The exact density is extended to the estimating equation (EE) approach and the nonlinear regression with an arbitrary number of linear parameters and one intrinsically nonlinear parameter. For a very special nonlinear regression model, the derived density coincides with the distribution of the ratio of two normally distributed random variables previously obtained by Fieller (1932), unlike other approximations previously suggested by other authors. Approximations to the density of the EE estimators are discussed in the multivariate case. Numerical complications associated with the nonlinear least squares are illustrated, such as nonexistence and/or multiple solutions, as major factors contributing to poor density approximation. The nonlinear Markov-Gauss theorem is formulated based on the near exact EE density approximation.

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

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

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

2015-10-15

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

Nonlinear evolution equations for surface plasmons for nano-focusing at a Kerr/metallic interface and tapered waveguide

NASA Astrophysics Data System (ADS)

Crutcher, Sihon H.; Osei, Albert; Biswas, Anjan

2012-06-01

Maxwell's equations for a metallic and nonlinear Kerr interface waveguide at the nanoscale can be approximated to a (1+1) D Nonlinear Schrodinger type model equation (NLSE) with appropriate assumptions and approximations. Theoretically, without losses or perturbations spatial plasmon solitons profiles are easily produced. However, with losses, the amplitude or beam profile is no longer stationary and adiabatic parameters have to be considered to understand propagation. For this model, adiabatic parameters are calculated considering losses resulting in linear differential coupled integral equations with constant definite integral coefficients not dependent on the transverse and longitudinal coordinates. Furthermore, by considering another configuration, a waveguide that is an M-NL-M (metal-nonlinear Kerr-metal) that tapers, the tapering can balance the loss experienced at a non-tapered metal/nonlinear Kerr interface causing attenuation of the beam profile, so these spatial plasmon solitons can be produced. In this paper taking into consideration the (1+1)D NLSE model for a tapered waveguide, we derive a one soliton solution based on He's Semi-Inverse Variational Principle (HPV).

On equations of motion of a nonlinear hydroelastic structure

NASA Astrophysics Data System (ADS)

Plotnikov, P. I.; Kuznetsov, I. V.

2008-07-01

Formal derivation of equations of a nonlinear hydroelastic structure, which is a volume of an ideal incompressible fluid covered by a shell, is proposed. The study is based on two assumptions. The first assumption implies that the energy stored in the shell is completely determined by the mean curvature and by the elementary area. In a three-dimensional case, the energy stored in the shell is chosen in the form of the Willmore functional. In a two-dimensional case, a more generic form of the functional can be considered. The second assumption implies that the equations of motionhave a Hamiltonian structure and can be obtained from the Lagrangian variational principle. In a two-dimensional case, a condition for the hydroelastic structure is derived, which relates the external pressure and the curvature of the elastic shell.

Discovering governing equations from data by sparse identification of nonlinear dynamical systems

PubMed Central

Brunton, Steven L.; Proctor, Joshua L.; Kutz, J. Nathan

2016-01-01

Extracting governing equations from data is a central challenge in many diverse areas of science and engineering. Data are abundant whereas models often remain elusive, as in climate science, neuroscience, ecology, finance, and epidemiology, to name only a few examples. In this work, we combine sparsity-promoting techniques and machine learning with nonlinear dynamical systems to discover governing equations from noisy measurement data. The only assumption about the structure of the model is that there are only a few important terms that govern the dynamics, so that the equations are sparse in the space of possible functions; this assumption holds for many physical systems in an appropriate basis. In particular, we use sparse regression to determine the fewest terms in the dynamic governing equations required to accurately represent the data. This results in parsimonious models that balance accuracy with model complexity to avoid overfitting. We demonstrate the algorithm on a wide range of problems, from simple canonical systems, including linear and nonlinear oscillators and the chaotic Lorenz system, to the fluid vortex shedding behind an obstacle. The fluid example illustrates the ability of this method to discover the underlying dynamics of a system that took experts in the community nearly 30 years to resolve. We also show that this method generalizes to parameterized systems and systems that are time-varying or have external forcing. PMID:27035946

Discovering governing equations from data by sparse identification of nonlinear dynamical systems.

PubMed

Brunton, Steven L; Proctor, Joshua L; Kutz, J Nathan

2016-04-12

Extracting governing equations from data is a central challenge in many diverse areas of science and engineering. Data are abundant whereas models often remain elusive, as in climate science, neuroscience, ecology, finance, and epidemiology, to name only a few examples. In this work, we combine sparsity-promoting techniques and machine learning with nonlinear dynamical systems to discover governing equations from noisy measurement data. The only assumption about the structure of the model is that there are only a few important terms that govern the dynamics, so that the equations are sparse in the space of possible functions; this assumption holds for many physical systems in an appropriate basis. In particular, we use sparse regression to determine the fewest terms in the dynamic governing equations required to accurately represent the data. This results in parsimonious models that balance accuracy with model complexity to avoid overfitting. We demonstrate the algorithm on a wide range of problems, from simple canonical systems, including linear and nonlinear oscillators and the chaotic Lorenz system, to the fluid vortex shedding behind an obstacle. The fluid example illustrates the ability of this method to discover the underlying dynamics of a system that took experts in the community nearly 30 years to resolve. We also show that this method generalizes to parameterized systems and systems that are time-varying or have external forcing.

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

PubMed

Liu, Wei; Zhang, Jing; Li, Xiliang

2018-01-01

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

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

PubMed Central

Zhang, Jing; Li, Xiliang

2018-01-01

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

Generalized nonlinear Schrödinger equation for dispersive susceptibility and permeability: application to negative index materials.

PubMed

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.

Nonlinear truncation error analysis of finite difference schemes for the Euler equations

NASA Technical Reports Server (NTRS)

Klopfer, G. H.; Mcrae, D. S.

1983-01-01

It is pointed out that, in general, dissipative finite difference integration schemes have been found to be quite robust when applied to the Euler equations of gas dynamics. The present investigation considers a modified equation analysis of both implicit and explicit finite difference techniques as applied to the Euler equations. The analysis is used to identify those error terms which contribute most to the observed solution errors. A technique for analytically removing the dominant error terms is demonstrated, resulting in a greatly improved solution for the explicit Lax-Wendroff schemes. It is shown that the nonlinear truncation errors are quite large and distributed quite differently for each of the three conservation equations as applied to a one-dimensional shock tube problem.

An efficient computational method for solving nonlinear stochastic Itô integral equations: Application for stochastic problems in physics

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

Heydari, M.H., E-mail: heydari@stu.yazd.ac.ir; The Laboratory of Quantum Information Processing, Yazd University, Yazd; Hooshmandasl, M.R., E-mail: hooshmandasl@yazd.ac.ir

Because of the nonlinearity, closed-form solutions of many important stochastic functional equations are virtually impossible to obtain. Thus, numerical solutions are a viable alternative. In this paper, a new computational method based on the generalized hat basis functions together with their stochastic operational matrix of Itô-integration is proposed for solving nonlinear stochastic Itô integral equations in large intervals. In the proposed method, a new technique for computing nonlinear terms in such problems is presented. The main advantage of the proposed method is that it transforms problems under consideration into nonlinear systems of algebraic equations which can be simply solved. Errormore » analysis of the proposed method is investigated and also the efficiency of this method is shown on some concrete examples. The obtained results reveal that the proposed method is very accurate and efficient. As two useful applications, the proposed method is applied to obtain approximate solutions of the stochastic population growth models and stochastic pendulum problem.« less

A mass-energy preserving Galerkin FEM for the coupled nonlinear fractional Schrödinger equations

NASA Astrophysics Data System (ADS)

Zhang, Guoyu; Huang, Chengming; Li, Meng

2018-04-01

We consider the numerical simulation of the coupled nonlinear space fractional Schrödinger equations. Based on the Galerkin finite element method in space and the Crank-Nicolson (CN) difference method in time, a fully discrete scheme is constructed. Firstly, we focus on a rigorous analysis of conservation laws for the discrete system. The definitions of discrete mass and energy here correspond with the original ones in physics. Then, we prove that the fully discrete system is uniquely solvable. Moreover, we consider the unconditionally convergent properties (that is to say, we complete the error estimates without any mesh ratio restriction). We derive L2-norm error estimates for the nonlinear equations and L^{∞}-norm error estimates for the linear equations. Finally, some numerical experiments are included showing results in agreement with the theoretical predictions.

A new class of exact, nonlinear solutions to the Grad-Shafranov equation

NASA Technical Reports Server (NTRS)

Roumeliotis, George

1993-01-01

We have constructed a new class of exact, nonlinear solutions to the Grad-Shafranov equation, representing force-free magnetic fields with translational symmetry. These exact solutions are pertinent to the study of magnetic structures in the solar corona that are subjected to photospheric shearing motions.

Rogue wave solutions for the infinite integrable nonlinear Schrödinger equation hierarchy.

PubMed

Ankiewicz, A; Akhmediev, N

2017-07-01

We present rogue wave solutions of the integrable nonlinear Schrödinger equation hierarchy with an infinite number of higher-order terms. The latter include higher-order dispersion and higher-order nonlinear terms. In particular, we derive the fundamental rogue wave solutions for all orders of the hierarchy, with exact expressions for velocities, phase, and "stretching factors" in the solutions. We also present several examples of exact solutions of second-order rogue waves, including rogue wave triplets.

Maximum Likelihood Estimation of Nonlinear Structural Equation Models with Ignorable Missing Data

ERIC Educational Resources Information Center

Lee, Sik-Yum; Song, Xin-Yuan; Lee, John C. K.

2003-01-01

The existing maximum likelihood theory and its computer software in structural equation modeling are established on the basis of linear relationships among latent variables with fully observed data. However, in social and behavioral sciences, nonlinear relationships among the latent variables are important for establishing more meaningful models…

Arbitrarily high-order time-stepping schemes based on the operator spectrum theory for high-dimensional nonlinear Klein-Gordon equations

NASA Astrophysics Data System (ADS)

Liu, Changying; Wu, Xinyuan

2017-07-01

In this paper we explore arbitrarily high-order Lagrange collocation-type time-stepping schemes for effectively solving high-dimensional nonlinear Klein-Gordon equations with different boundary conditions. We begin with one-dimensional periodic boundary problems and first formulate an abstract ordinary differential equation (ODE) on a suitable infinity-dimensional function space based on the operator spectrum theory. We then introduce an operator-variation-of-constants formula which is essential for the derivation of our arbitrarily high-order Lagrange collocation-type time-stepping schemes for the nonlinear abstract ODE. The nonlinear stability and convergence are rigorously analysed once the spatial differential operator is approximated by an appropriate positive semi-definite matrix under some suitable smoothness assumptions. With regard to the two dimensional Dirichlet or Neumann boundary problems, our new time-stepping schemes coupled with discrete Fast Sine / Cosine Transformation can be applied to simulate the two-dimensional nonlinear Klein-Gordon equations effectively. All essential features of the methodology are present in one-dimensional and two-dimensional cases, although the schemes to be analysed lend themselves with equal to higher-dimensional case. The numerical simulation is implemented and the numerical results clearly demonstrate the advantage and effectiveness of our new schemes in comparison with the existing numerical methods for solving nonlinear Klein-Gordon equations in the literature.

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

NASA Astrophysics Data System (ADS)

Vitanov, Nikolay K.

2011-03-01

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

Exact Solutions of Atmospheric (2+1)-Dimensional Nonlinear Incompressible Non-hydrostatic Boussinesq Equations

NASA Astrophysics Data System (ADS)

Liu, Ping; Wang, Ya-Xiong; Ren, Bo; Li, Jin-Hua

2016-12-01

Exact solutions of the atmospheric (2+1)-dimensional nonlinear incompressible non-hydrostatic Boussinesq (INHB) equations are researched by Combining function expansion and symmetry method. By function expansion, several expansion coefficient equations are derived. Symmetries and similarity solutions are researched in order to obtain exact solutions of the INHB equations. Three types of symmetry reduction equations and similarity solutions for the expansion coefficient equations are proposed. Non-traveling wave solutions for the INHB equations are obtained by symmetries of the expansion coefficient equations. Making traveling wave transformations on expansion coefficient equations, we demonstrate some traveling wave solutions of the INHB equations. The evolutions on the wind velocities, temperature perturbation and pressure perturbation are demonstrated by figures, which demonstrate the periodic evolutions with time and space. Supported by the National Natural Science Foundation of China under Grant Nos. 11305031 and 11305106, and Training Programme Foundation for Outstanding Young Teachers in Higher Education Institutions of Guangdong Province under Grant No. Yq2013205

On a new class of completely integrable nonlinear wave equations. II. Multi-Hamiltonian structure

NASA Astrophysics Data System (ADS)

Nutku, Y.

1987-11-01

The multi-Hamiltonian structure of a class of nonlinear wave equations governing the propagation of finite amplitude waves is discussed. Infinitely many conservation laws had earlier been obtained for these equations. Starting from a (primary) Hamiltonian formulation of these equations the necessary and sufficient conditions for the existence of bi-Hamiltonian structure are obtained and it is shown that the second Hamiltonian operator can be constructed solely through a knowledge of the first Hamiltonian function. The recursion operator which first appears at the level of bi-Hamiltonian structure gives rise to an infinite sequence of conserved Hamiltonians. It is found that in general there exist two different infinite sequences of conserved quantities for these equations. The recursion relation defining higher Hamiltonian structures enables one to obtain the necessary and sufficient conditions for the existence of the (k+1)st Hamiltonian operator which depends on the kth Hamiltonian function. The infinite sequence of conserved Hamiltonians are common to all the higher Hamiltonian structures. The equations of gas dynamics are discussed as an illustration of this formalism and it is shown that in general they admit tri-Hamiltonian structure with two distinct infinite sets of conserved quantities. The isothermal case of γ=1 is an exceptional one that requires separate treatment. This corresponds to a specialization of the equations governing the expansion of plasma into vacuum which will be shown to be equivalent to Poisson's equation in nonlinear acoustics.

Solitonic dynamics and excitations of the nonlinear Schrödinger equation with third-order dispersion in non-Hermitian PT-symmetric potentials.

PubMed

Chen, Yong; Yan, Zhenya

2016-03-22

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.

Solitonic dynamics and excitations of the nonlinear Schrödinger equation with third-order dispersion in non-Hermitian PT-symmetric potentials

PubMed Central

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

A homotopy analysis method for the nonlinear partial differential equations arising in engineering

NASA Astrophysics Data System (ADS)

Hariharan, G.

2017-05-01

In this article, we have established the homotopy analysis method (HAM) for solving a few partial differential equations arising in engineering. This technique provides the solutions in rapid convergence series with computable terms for the problems with high degree of nonlinear terms appearing in the governing differential equations. The convergence analysis of the proposed method is also discussed. Finally, we have given some illustrative examples to demonstrate the validity and applicability of the proposed method.

The solution of non-linear hyperbolic equation systems by the finite element method

NASA Technical Reports Server (NTRS)

Loehner, R.; Morgan, K.; Zienkiewicz, O. C.

1984-01-01

A finite-element method for the solution of nonlinear hyperbolic systems of equations, such as those encountered in non-self-adjoint problems of transient phenomena in convection-diffusion or in the mixed representation of wave problems, is developed and demonstrated. The problem is rewritten in moving coordinates and reinterpolated to the original mesh by a Taylor expansion prior to a standard Galerkin spatial discretization, and it is shown that this procedure is equivalent to the time-discretization approach of Donea (1984). Numerical results for sample problems are presented graphically, including such shallow-water problems as the breaking of a dam, the shoaling of a wave, and the outflow of a river; compressible flows such as the isothermal flow in a nozzle and the Riemann shock-tube problem; and the two-dimensional scalar-advection, nonlinear-shallow-water, and Euler equations.

Exploring inductive linearization for pharmacokinetic-pharmacodynamic systems of nonlinear ordinary differential equations.

PubMed

Hasegawa, Chihiro; Duffull, Stephen B

2018-02-01

Pharmacokinetic-pharmacodynamic systems are often expressed with nonlinear ordinary differential equations (ODEs). While there are numerous methods to solve such ODEs these methods generally rely on time-stepping solutions (e.g. Runge-Kutta) which need to be matched to the characteristics of the problem at hand. The primary aim of this study was to explore the performance of an inductive approximation which iteratively converts nonlinear ODEs to linear time-varying systems which can then be solved algebraically or numerically. The inductive approximation is applied to three examples, a simple nonlinear pharmacokinetic model with Michaelis-Menten elimination (E1), an integrated glucose-insulin model and an HIV viral load model with recursive feedback systems (E2 and E3, respectively). The secondary aim of this study was to explore the potential advantages of analytically solving linearized ODEs with two examples, again E3 with stiff differential equations and a turnover model of luteinizing hormone with a surge function (E4). The inductive linearization coupled with a matrix exponential solution provided accurate predictions for all examples with comparable solution time to the matched time-stepping solutions for nonlinear ODEs. The time-stepping solutions however did not perform well for E4, particularly when the surge was approximated by a square wave. In circumstances when either a linear ODE is particularly desirable or the uncertainty in matching the integrator to the ODE system is of potential risk, then the inductive approximation method coupled with an analytical integration method would be an appropriate alternative.

A New Family of Schroder's Method and Its Variants Based on Power Means for Multiple Roots of Nonlinear Equations

ERIC Educational Resources Information Center

Kanwar, V.; Sharma, Kapil K.; Behl, Ramandeep

2010-01-01

In this article, we derive one-parameter family of Schroder's method based on Gupta et al.'s (K.C. Gupta, V. Kanwar, and S. Kumar, "A family of ellipse methods for solving non-linear equations", Int. J. Math. Educ. Sci. Technol. 40 (2009), pp. 571-575) family of ellipse methods for the solution of nonlinear equations. Further, we introduce new…

Additivity of nonlinear biomass equations

Treesearch

Bernard R. Parresol

2001-01-01

Two procedures that guarantee the property of additivity among the components of tree biomass and total tree biomass utilizing nonlinear functions are developed. Procedure 1 is a simple combination approach, and procedure 2 is based on nonlinear joint-generalized regression (nonlinear seemingly unrelated regressions) with parameter restrictions. Statistical theory is...

Parametric reduced models for the nonlinear Schrödinger equation

NASA Astrophysics Data System (ADS)

Harlim, John; Li, Xiantao

2015-05-01

Reduced models for the (defocusing) nonlinear Schrödinger equation are developed. In particular, we develop reduced models that only involve the low-frequency modes given noisy observations of these modes. The ansatz of the reduced parametric models are obtained by employing a rational approximation and a colored-noise approximation, respectively, on the memory terms and the random noise of a generalized Langevin equation that is derived from the standard Mori-Zwanzig formalism. The parameters in the resulting reduced models are inferred from noisy observations with a recently developed ensemble Kalman filter-based parametrization method. The forecasting skill across different temperature regimes are verified by comparing the moments up to order four, a two-time correlation function statistics, and marginal densities of the coarse-grained variables.

Parametric reduced models for the nonlinear Schrödinger equation.

PubMed

Harlim, John; Li, Xiantao

2015-05-01

Reduced models for the (defocusing) nonlinear Schrödinger equation are developed. In particular, we develop reduced models that only involve the low-frequency modes given noisy observations of these modes. The ansatz of the reduced parametric models are obtained by employing a rational approximation and a colored-noise approximation, respectively, on the memory terms and the random noise of a generalized Langevin equation that is derived from the standard Mori-Zwanzig formalism. The parameters in the resulting reduced models are inferred from noisy observations with a recently developed ensemble Kalman filter-based parametrization method. The forecasting skill across different temperature regimes are verified by comparing the moments up to order four, a two-time correlation function statistics, and marginal densities of the coarse-grained variables.

Symmetry-Breaking Bifurcation in the Nonlinear Schrödinger Equation with Symmetric Potentials

NASA Astrophysics Data System (ADS)

Kirr, E.; Kevrekidis, P. G.; Pelinovsky, D. E.

2011-12-01

We consider the focusing (attractive) nonlinear Schrödinger (NLS) equation with an external, symmetric potential which vanishes at infinity and supports a linear bound state. We prove that the symmetric, nonlinear ground states must undergo a symmetry breaking bifurcation if the potential has a non-degenerate local maxima at zero. Under a generic assumption we show that the bifurcation is either a subcritical or supercritical pitchfork. In the particular case of double-well potentials with large separation, the power of nonlinearity determines the subcritical or supercritical character of the bifurcation. The results are obtained from a careful analysis of the spectral properties of the ground states at both small and large values for the corresponding eigenvalue parameter.

Sqeezing generated by a nonlinear master equation and by amplifying-dissipative Hamiltonians

NASA Technical Reports Server (NTRS)

Dodonov, V. V.; Marchiolli, M. A.; Mizrahi, Solomon S.; Moussa, M. H. Y.

1994-01-01

In the first part of this contribution we show that the master equation derived from the generalized version of the nonlinear Doebner-Goldin equation leads to the squeezing of one of the quadratures. In the second part we consider two familiar Hamiltonians, the Bateman- Caldirola-Kanai and the optical parametric oscillator; going back to their classical Lagrangian form we introduce a stochastic force and a dissipative factor. From this new Lagrangian we obtain a modified Hamiltonian that treats adequately the simultaneous amplification and dissipation phenomena, presenting squeezing, too.

Multiple positive solutions to a coupled systems of nonlinear fractional differential equations.

PubMed

Shah, Kamal; Khan, Rahmat Ali

2016-01-01

In this article, we study existence, uniqueness and nonexistence of positive solution to a highly nonlinear coupled system of fractional order differential equations. Necessary and sufficient conditions for the existence and uniqueness of positive solution are developed by using Perov's fixed point theorem for the considered problem. Further, we also established sufficient conditions for existence of multiplicity results for positive solutions. Also, we developed some conditions under which the considered coupled system of fractional order differential equations has no positive solution. Appropriate examples are also provided which demonstrate our results.

Robust fast controller design via nonlinear fractional differential equations.

PubMed

Zhou, Xi; Wei, Yiheng; Liang, Shu; Wang, Yong

2017-07-01

A new method for linear system controller design is proposed whereby the closed-loop system achieves both robustness and fast response. The robustness performance considered here means the damping ratio of closed-loop system can keep its desired value under system parameter perturbation, while the fast response, represented by rise time of system output, can be improved by tuning the controller parameter. We exploit techniques from both the nonlinear systems control and the fractional order systems control to derive a novel nonlinear fractional order controller. For theoretical analysis of the closed-loop system performance, two comparison theorems are developed for a class of fractional differential equations. Moreover, the rise time of the closed-loop system can be estimated, which facilitates our controller design to satisfy the fast response performance and maintain the robustness. Finally, numerical examples are given to illustrate the effectiveness of our methods. Copyright © 2017 ISA. Published by Elsevier Ltd. All rights reserved.

Solution to the nonlinear field equations of ten dimensional supersymmetric Yang-Mills theory

NASA Astrophysics Data System (ADS)

Mafra, Carlos R.; Schlotterer, Oliver

2015-09-01

In this paper, we present a formal solution to the nonlinear field equations of ten-dimensional super Yang-Mills theory. It is assembled from products of linearized superfields which have been introduced as multiparticle superfields in the context of superstring perturbation theory. Their explicit form follows recursively from the conformal field theory description of the gluon multiplet in the pure spinor superstring. Furthermore, superfields of higher-mass dimensions are defined and their equations of motion are spelled out.

Soliton interactions and complexes for coupled nonlinear Schrödinger equations.

PubMed

Jiang, Yan; Tian, Bo; Liu, Wen-Jun; Sun, Kun; Li, Min; Wang, Pan

2012-03-01

Under investigation in this paper are the coupled nonlinear Schrödinger (CNLS) equations, which can be used to govern the optical-soliton propagation and interaction in such optical media as the multimode fibers, fiber arrays, and birefringent fibers. By taking the 3-CNLS equations as an example for the N-CNLS ones (N≥3), we derive the analytic mixed-type two- and three-soliton solutions in more general forms than those obtained in the previous studies with the Hirota method and symbolic computation. With the choice of parameters for those soliton solutions, soliton interactions and complexes are investigated through the asymptotic and graphic analysis. Soliton interactions and complexes with the bound dark solitons in a mode or two modes are observed, including that (i) the two bright solitons display the breatherlike structures while the two dark ones stay parallel, (ii) the two bright and dark solitons all stay parallel, and (iii) the states of the bound solitons change from the breatherlike structures to the parallel one even with the distance between those solitons smaller than that before the interaction with the regular one soliton. Asymptotic analysis is also used to investigate the elastic and inelastic interactions between the bound solitons and the regular one soliton. Furthermore, some discussions are extended to the N-CNLS equations (N>3). Our results might be helpful in such applications as the soliton switch, optical computing, and soliton amplification in the nonlinear optics.

An effective solution to the nonlinear, nonstationary Navier-Stokes equations for two dimensions

NASA Technical Reports Server (NTRS)

Gabrielsen, R. E.

1975-01-01

A sequence of approximate solutions for the nonlinear, nonstationary Navier-Stokes equations for a two-dimensional domain, from which explicit error estimates and rates of convergence are obtained, is described. This sequence of approximate solutions is based primarily on the Newton-Kantorovich method.

Newton's method for nonlinear stochastic wave equations driven by one-dimensional Brownian motion.

PubMed

Leszczynski, Henryk; Wrzosek, Monika

2017-02-01

We consider nonlinear stochastic wave equations driven by one-dimensional white noise with respect to time. The existence of solutions is proved by means of Picard iterations. Next we apply Newton's method. Moreover, a second-order convergence in a probabilistic sense is demonstrated.

The stability of full dimensional KAM tori for nonlinear Schrödinger equation

NASA Astrophysics Data System (ADS)

Cong, Hongzi; Liu, Jianjun; Shi, Yunfeng; Yuan, Xiaoping

2018-04-01

In this paper, it is proved that the full dimensional invariant tori obtained by Bourgain [J. Funct. Anal., 229 (2005), no. 1, 62-94] is stable in a very long time for 1D nonlinear Schrödinger equation with periodic boundary conditions.

Fourth order Douglas implicit scheme for solving three dimension reaction diffusion equation with non-linear source term

NASA Astrophysics Data System (ADS)

Hasnain, Shahid; Saqib, Muhammad; Mashat, Daoud Suleiman

2017-07-01

This research paper represents a numerical approximation to non-linear three dimension reaction diffusion equation with non-linear source term from population genetics. Since various initial and boundary value problems exist in three dimension reaction diffusion phenomena, which are studied numerically by different numerical methods, here we use finite difference schemes (Alternating Direction Implicit and Fourth Order Douglas Implicit) to approximate the solution. Accuracy is studied in term of L2, L∞ and relative error norms by random selected grids along time levels for comparison with analytical results. The test example demonstrates the accuracy, efficiency and versatility of the proposed schemes. Numerical results showed that Fourth Order Douglas Implicit scheme is very efficient and reliable for solving 3-D non-linear reaction diffusion equation.

Periodic solutions of second-order nonlinear difference equations containing a small parameter. II - Equivalent linearization

NASA Technical Reports Server (NTRS)

Mickens, R. E.

1985-01-01

The classical method of equivalent linearization is extended to a particular class of nonlinear difference equations. It is shown that the method can be used to obtain an approximation of the periodic solutions of these equations. In particular, the parameters of the limit cycle and the limit points can be determined. Three examples illustrating the method are presented.