Nonlinear Physical Equations Research Articles

Quasilinearization method: Nonperturbative approach to physical problems

The general properties of the quasilinearization method (QLM), particularly its fast quadratic convergence, monotonicity, and numerical stability, are analyzed and illustrated on different physical problems. The method approaches the solution of a nonlinear differential equation by approximating the nonlinear terms by a sequence of linear ones and is not based on the existence of a small parameter. It is shown that QLM gives excellent results when applied to different nonlinear differential equations in physics, such as Blasius, Lane-Emden, and Thomas-Fermi equations, as well as in computation of ground and excited bound-state energies and wave functions in quantum mechanics (where it can be applied by casting the Schrodinger equation in the nonlinear Riccati form) for a variety of potentials most of which are not treatable with the help of perturbation theory. The convergence of the QLM expansion of both energies and wave functions for all states is very fast and the first few iterations already yield extremely precise results. The QLM approximations, unlike the asymptotic series in perturbation theory and 1/N expansions, are not divergent at higher orders. The method sums many orders of perturbation theory as well as of the WKB expansion. It provides final and accurate answers for large and infinite values of the coupling constants and is able to handle even supersingular potentials for which each term of the perturbation series is infinite and the perturbation expansion does not exist.

A modified hyperbolic function method and its application to the Toda lattice

In this paper, based on the computer symbolic system--Maple, we presented a modified hyperbolic function method to search for the exact solutions of nonlinear differential-difference equations (DDEs). The discrete Toda lattice is chosen to illustrate the method, and abundant new exact solutions are obtained, which include kink-shaped solitary wave solutions and other new types of exact solutions. This method can also be applied to many other nonlinear differential-difference equations (DDEs) in mathematics physics.

A new Riccati equation expansion method with symbolic computation to construct new travelling wave solution of nonlinear differential equations

Abstract In this paper, based on a new system of Riccati equations, we present a triple Riccati equation expansion method. The nonlinear Klein–Gordon equation and the combined KdV–mKdV equation are chosen to illustrate the method. As a result, many new travelling solutions, which include bell-profile solitary wave solution, singular solitary solutions, periodic solutions are obtained. This method can be used to the other nonlinear evolution equations in mathematics physics.

Jacobi Elliptic Function Solutions of Three Coupled Nonlinear Physical Equations

Abstract The Jacobi elliptic function solutions of coupled nonlinear partial differential equations, including the coupled modified KdV (mKdV) equations, long-short-wave interaction system and the Davey- Stewartson (DS) equations, are obtained by using the mixed dn-sn method. The solutions obtained in this paper include the single and the combined Jacobi elliptic function solutions. In the limiting case, the solitary wave solutions of the systems are also given. - PACS: 02.30.Jr; 03.40.Kf; 03.65.Fd

Exact solutions to two coupled nonlinear physical equations

In this paper, a two-step ansatz is proposed, which leads to some new solutions to two coupled nonlinear physical equations.

Weak transversality and partially invariant solutions

New exact solutions are obtained for several nonlinear physical equations, namely the Navier–Stokes and Euler systems, an isentropic compressible fluid system and a vector nonlinear Schrödinger equation. The solution method makes use of the symmetry group of the system in situations when the standard Lie method of symmetry reduction is not applicable.

Quasilinearization approach to nonlinear problems in physics with application to nonlinear ODEs

The general conditions under which the quadratic, uniform and monotonic convergence in the quasilinearization method of solving nonlinear ordinary differential equations could be proved are formulated and elaborated. The generalization of the proof to partial differential equations is straight forward. The method, whose mathematical basis in physics was discussed recently by one of the present authors (VBM), approximates the solution of a nonlinear differential equation by treating the nonlinear terms as a perturbation about the linear ones, and unlike perturbation theories is not based on the existence of some kind of a small parameter. It is shown that the quasilinearization method gives excellent results when applied to different nonlinear ordinary differential equations in physics, such as the Blasius, Duffing, Lane–Emden and Thomas–Fermi equations. The first few quasilinear iterations already provide extremely accurate and numerically stable answers.

Explicit solutions to three nonlinear physical models

Explicit solutions are derived for three coupled nonlinear physical model equations by using a delicate way of ansatz method.

CONNECTIONS AMONG HOMOGENEOUS BALANCE METHOD, WEISS-TABOR-CARNEVALE METHOD AND CLARKSON-KRUSKAL METHOD

The homogeneous balance method is extended to search for the Bcklund transformations and similarity reductions of nonlinear mathematical and physical equations. The corresponding results coincide with those of Weiss-Tabor-Carnevale method and Clarkson-Kruskal method respectively. It is shown that there exist close relations among these three methods.

Nonlinear equations in soliton physics and operator ideals

An operator-theoretic method for the investigation of nonlinear equations in soliton physics is discussed comprehensively. Originating from pioneering work of Marchenko, our operator-method is based on new insights into the theory of traces and determinants on operator ideals. Therefore, we give a systematic and concise approach to some recent developments in this direction which are important in the context of this paper. Our method is widely applicable. We carry out the corresponding arguments in detail for the Kadomtsev-Petviashvili equation and summarize the results concerning the Korteweg-de Vries and the modified Korteweg-de Vries equation as well as for the sine-Gordon equation. Exactly the same formalism works in the discrete case, as the treatment of the Toda lattice, the Langmuir and the Wadati lattice shows. AMS classification scheme numbers: 35C05, 35Q51, 35Q53, 35Q58, 47D50, 47N20

Exact and explicit solitary wave solutions to some nonlinear equations

Exact and explicit solitary wave solutions are obtained for some physically interesting nonlinear evolutions and wave equations in physics and other fields by using a special transformation. These equations include the KdV-Burgers equation, the MKdV-Burgers equation, the combined KdV-MKdV equation, the Newell-Whitehead equation, the dissipative Φ4-model equation, the generalized Fisher equation, and the elastic-medium wave equation.

On realizations of solutions of the KdV equation by determinants on operator ideals

Using new developments in the theory of traces, determinants, and elementary operators on quasi-Banach operator ideals we clarify and extend Marchenko’s method for realizing solutions of the KdV equation. Moreover, we point out why abstract traces and determinants on quasi-Banach operator ideals are appropriate tools for obtaining solutions of the KdV equation. The method we present can also be applied to other nonlinear equations in soliton physics.

Exact wave front solutions to two generalized coupled nonlinear physical equations

We find the analytical wave front solutions to two coupled physical models by presenting various ansatze for the two unknowns in the equations of interest.

Predicting the transverse strength of unidirectional composites under combination loading

This article presents a mathematical model for predicting the transverse strength of unidirectional fiber composites subjected to combination transverse loading under different conditions. The behavior of the matrix is described by nonlinear physical equations consistent with the strain theory of plasticity for the active loading section. The fibers are assumed to be isotropic and elastic. The boundary-value problem of micromechanics that is formulated includes strength criteria for the matrix and fibers that mark the beginning of their possible failure. The modeling of the fracture process is taken farther through the use of a scheme that reduces the stiffness of the matrix and fibers in the failed regions in relation to the sign of the first invariant of the stress tensor. The method of local approximation is used together with the finite-element method to calculate the stress and strain fields in unidirectional composites with cylindrical fibers in a tetragonal layup. The model is used to study the behavior of an epoxy-based organic-fiber-reinforced plastic subjected to transverse loading in different simple paths — including simultaneous compressive and tensile loads, as well as transverse shear.

An investigation of the asymptotic decomposition method for nonlinear equations in physics

This paper considers an interesting variation of the author's decomposition method, first proposed in 1983, which permits determination of the behavior of the solution of a differential equation as the independent variable approaches infinity. Extensions to partial differential equations are also considered.

Non-linear partial differential equations in physics. II. Recent advances

For pt.I, see ibid. vol.1, no.3, p.162 (1980). Non-linear partial differential equations of physics are discussed using water waves in a channel as an illustration. Weakly non-linear waves, wave breaking, shocks and solitary waves have already been considered in part I. Recent progress on the interaction of solitary waves, solitons, and the inverse scattering method for solving non-linear partial differential equations is described.

Non-linear partial differential equations in physics. I. Earlier progress

Non-linear partial differential equations in physics are discussed using water waves in a channel as an illustration. Weakly non-linear waves, wave breaking shocks, and solitary waves are considered. Although these topics have been understood for many years, they are unfamiliar to the majority of physicists. They are also the background to the recent advances in non-linear physics which will be discussed in a companion paper.

Nonlinear Equations in Physics and Mathematics – Proceedings of the Nato Advanced Study Institute held in Istanbul 1–13 August 1977

A O Barut (ed) 1978 Dordrecht, Holland: D Reidel viii + 473 pp price $39 The scope of this book is given inside the dust jacket as: classical solvable nonlinear evolution equations; integral dynamical systems; inverse scattering method; solitons and their occurrence in physics; nonlinear equations in lattices and in continuum physics; and applications to quantum field theory and elementary particles. The book will no doubt be helpful to graduate students and researchers in mathematics and physics where analytical solutions are being soughtbut, by their very nature, the lectures are brief introductions to very specialised topics.

Nonlinear Stochastic Differential Equations in Statistical Physics and Integral Representation

Vol. 56 (1976), No. 6, next sheet to page 2001, line 11. The volume number was misprinted clue to a technical mistake. “Volume 55” should read “Volume 56”. The correct sheet is attached in the following page. Publication Office

Nonlinear Stochastic Differential Equations in Statistical Physics and Integral Representation

Integral representation (IR), or the so-called ·wiener-Hermite expansion, is proposed as a new method of solving a class of nonlinear stochastic differential equations (NLSDE's) that appear in the theory of Brownian motion of anharmonic oscillators, semi-classical treatment of laser oscillation or in the Kraichnan-Wyld formulation of turbulence. The basic procedure of application of IR is established and simple NLSDE's are analyzed. Results for power spectral density agree well with numerical experiment data even in strongly nonlinear cases. It is also shown that IR is a completed form of renormalized perturbation expansion, and provides a systematic way to improve various approximation schemes such as fourth cumulant discard approximation or equivalent linearization procedure.