Asymmetric Discrete Envelope Soliton Solutions for Semi-Discrete Nonlinear Wave Equations

In this paper, the soliton solutions and the corresponding conservation laws of a few nonlinear wave equations will be obtained. The Hunter-Saxton equation, the improved Korteweg-de Vries equation, and other such equations will be considered. The Lie symmetry approach will be utilized to extract the conserved densities of these equations. The soliton solutions will be used to obtain the conserved quantities of these equations.

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EXACT SOLUTIONS AND THEIR DYNAMICS OF TRAVELING WAVES IN THREE TYPICAL NONLINEAR WAVE EQUATIONS

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127409024049 ◽

2009 ◽

Vol 19 (07) ◽

pp. 2249-2266 ◽

Cited By ~ 12

Author(s):

JIBIN LI ◽

YI ZHANG ◽

GUANRONG CHEN

Keyword(s):

Nonlinear Wave ◽

Traveling Waves ◽

Traveling Wave ◽

Wave Equations ◽

Visual Illusion ◽

Soliton Solutions ◽

Traveling Wave Solutions ◽

Dynamical Analysis ◽

Nonlinear Wave Equations ◽

Wave Solutions

It was reported in the literature that some nonlinear wave equations have the so-called loop- and inverted-loop-soliton solutions, as well as the so-called loop-periodic solutions. Are these true mathematical solutions or just numerical artifacts? To answer the question, this article investigates all traveling wave solutions in the parameter space for three typical nonlinear wave equations from a theoretical viewpoint of dynamical systems. Dynamical analysis shows that all these loop- and inverted-loop-solutions are merely visual illusion of numerical artifacts. To reveal the nature of such special phenomena, this article also offers the mathematical parametric representations of these traveling wave solutions precisely in analytic forms.

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A REDUCTION mKdV METHOD WITH SYMBOLIC COMPUTATION TO CONSTRUCT NEW DOUBLY-PERIODIC SOLUTIONS FOR NONLINEAR WAVE EQUATIONS

International Journal of Modern Physics C ◽

10.1142/s0129183103004814 ◽

2003 ◽

Vol 14 (05) ◽

pp. 661-672 ◽

Cited By ~ 12

Author(s):

ZHENYA YAN

Keyword(s):

Periodic Solutions ◽

Nonlinear Wave ◽

Symbolic Computation ◽

Wave Equations ◽

Kdv Equation ◽

Soliton Solutions ◽

Transformation Method ◽

Mkdv Equation ◽

Nonlinear Wave Equations ◽

Cubic Nonlinear Schrödinger Equation

Firstly twenty-four types of doubly-periodic solutions of the reduction mKdV equation are given. Secondly based on the reduction mKdV equation and its solutions, a systemic transformation method (called the reduction mKdV method) is developed to construct new doubly-periodic solutions of nonlinear equations. Thirdly with the aid of symbolic computation, we choose the KdV equation, the coupled variant Boussinesq equation and the cubic nonlinear Schrödinger equation to illustrate our method. As a result many types of solutions are obtained. These show that this method is simple and powerful to obtain more exact solutions including doubly-periodic solutions, soliton solutions and singly-periodic solutions to a wide class of nonlinear wave equations. Finally we further extended the method to a general form.

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