一类广义非线性Schroedinger方程的孤立子解及其性质研究

研究一类广义非线性Ｓｃｈｒｏｒｄｉｎｇｅｒ方程的孤立子解及其性质，研究非线性参数变化时孤立子性态的变化规律，同时研究该系统的数值解法，得到了一类广义非线性Ｓｃｈｒｏｅｄｉｎｇｅｒ方程差分格式的收敛性和稳定性的条件。     

耦合KdV方程的约化及求解

利用Clarkson和Kruskal(CK)直接方法,对耦合KdV方程进行相似约化,同时从李群出发对该约化方程作了群论解释.进一步地,借助Ablowitz-Ramani-Segur(ARS)算法对耦合方程展开Painlevé测试,找到了3个Painlevé可积模型.最后通过形变映射法,求得耦合KdV方程的准确解析解. 关键词： 耦合KdV方程 CK直接法 Painlevé分析法 准确解析解     

孤立子及其相互作用的初等分析

用较简捷而直观的试探解法几种典型的孤立子方程（ＫｄＶ方程，Ｔｏｄａ晶格波方程，ＳＧ方程与ＮＬＳ方程）求得单孤子解和两孤子解，以说明孤子及孤立子间相互作用的特点，并讨论了物理意义。     

孤立子简介

 早在1834年8月,26岁的英国科学家、造船工程师约翰·斯各脱·罗素在勘探连接爱丁堡和格拉斯哥的运河河道中偶然发现了一种十分奇妙的自然现象.1844年,他在《英国科学促进协会第14届会议报告》上发表了“论波动”一文,对这一现象作了极为生动的描述:“1834年秋,我观察过一条船被两匹马拉着沿狭窄的运河迅速前进.突然,船停了下来.     

KdV方程的左行孤子解及其相互作用

本文指出,KdV方程允许存在左行孤子、静态孤子和左、右行孤子的相互作用解。左行孤子的振幅越大,运动得越慢。 关键词：     

用影射法求解KdV方程

应用影射法求解KdV方程,得到了KdV方程的物理上有意义的两种类型的行波解,即孤立波解和Jacobi椭圆函数解,并绘图加以说明.     

组合KdV方程的双扭结孤立波及其稳定性研究

利用扩展的双曲函数法得到了combined KdV-mKdV (cKdV)方程的几类精确解,其中一类为具有扭结—反扭结状结构的双扭结单孤子解.在不同的极限情况下,该解分别退化为cKdV方程的扭结状或钟状孤波解.理论分析表明,cKdV方程既有传播型孤立波解,也有非传播型孤立波解.文中对双扭结型孤立波解的稳定性进行了数值研究,结果表明,cKdV方程既存在稳定的双扭结型孤立波,也存在不稳定的双扭结型孤立波. 关键词： cKdV方程 双扭结单孤子 稳定性     

耦合Burgers方程的对称性及对称约化

对称性分析是自然科学研究中的重要方法之一. 利用对称性分析研究了一个描述两层流体体系的模型即耦合Burgers方程的对称性. 利用对称性给出了这个模型的四种对称性约化并给出了这些约化方程的一些特殊的严格解,如有理解、行波孤立子解和非行波孤立子解. 关键词： 对称性约化 耦合Burgers方程 孤立子     

离散修正KdV方程的解析近似解

将同伦分析法进行了推广,使之适用于求解离散修正KdV方程.获得了由指数函数表达的亮孤子解,该解析近似解与精确解符合很好.数值模拟结果说明了同伦分析法对求解复杂非线性问题的有效性和潜力.     

mKdV方程的双扭结单孤子及其稳定性研究

基于双曲函数法的思想,通过选择新的展开函数,得到了modified Korteweg-de Vries(mKdV)方程的几类精确解,其中一类为具有扭结—反扭结状结构的双扭结单孤子解.在不同的极限情况下,该解分别退化为mKdV方程的扭结状或钟状孤波解.文中对双扭结型孤子解的稳定性进行了数值研究,结果表明:在长波和短波简谐波扰动、钟型孤立波扰动与随机扰动下,该孤子均稳定.     

Approximate analytic solutions for a generalized Hirota—Satsuma coupled KdV equation and a coupled mKdV equation

<正>This paper applies the variational iteration method to obtain approximate analytic solutions of a generalized Hirota-Satsuma coupled Korteweg-de Vries(KdV) equation and a coupled modified Korteweg-de Vries(mKdV) equation. This method provides a sequence of functions which converges to the exact solution of the problem and is based on the use of the Lagrange multiplier for the identification of optimal values of parameters in a functional.Some examples are given to demonstrate the reliability and convenience of the method and comparisons are made with the exact solutions.     

The extended auxiliary equation method for the KdV equation with variable coefficients

This paper applies an extended auxiliary equation method to obtain exact solutions of the KdV equation with variable coefficients. As a result, solitary wave solutions, trigonometric function solutions, rational function solutions, Jacobi elliptic doubly periodic wave solutions, and nonsymmetrical kink solution are obtained. It is shown that the extended auxiliary equation method, with the help of a computer symbolic computation system, is reliable and effective in finding exact solutions of variable coefficient nonlinear evolution equations in mathematical physics.     

The Jacobi elliptic function-like exact solutions to two kinds of KdV equations with variable coefficients and KdV equation with forcible term

By use of an auxiliary equation and through a function transformation, the Jacobi elliptic function wave-like solutions, the degenerated soliton-like solutions and the triangle function wave solutions to two kinds of Korteweg--de Vries (KdV) equations with variable coefficients and a KdV equation with a forcible term are constructed with the help of symbolic computation system Mathematica, where the new solutions are also constructed.     

Exact traveling wave solutions for a generalized Hirota-Satsuma coupled KdV equation by Fan sub-equation method

In this Letter, the Fan sub-equation method is used to construct exact solutions of a generalized Hirota-Satsuma coupled KdV equation. Many exact traveling wave solutions are successfully obtained, which contain more general solitary wave solutions and Jacobian elliptic function solutions with double periods. This method is straightforward and concise, and it can also be applied to other nonlinear evolution equations.     

Linear superposition of Wronskian rational solutions to the KdV equation

A linear superposition is studied for Wronskian rational solutions to the Kd V equation, which include rogue wave solutions. It is proved that it is equivalent to a polynomial identity that an arbitrary linear combination of two Wronskian polynomial solutions with a difference two between the Wronskian orders is again a solution to the bilinear Kd V equation. It is also conjectured that there is no other rational solutions among general linear superpositions of Wronskian rational solutions.     

The novel multi-solitary wave solution to the fifth-order KdV equation

By using Hirota‘s method, the novel multi-solitary wave solutions to the fifth-order KdV equation are obtained. Furthermore, various new solitary wave solutions are also derived by a reconstructed bilinear Bǎcklund transformation.     

修正cKdV方程组的孤立波结构及其稳定性

利用函数展开法求解修正耦合KdV(Coupled KdV,cKdV)方程组,得到几类孤立波解,包括扭结型-钟型、双扭结型、双钟型以及双扭结-双钟型结构的单孤子解.在不同的极限情况下,这些解分别退化为修正cKdV方程的扭结状或钟状孤波解.对孤立波的稳定性进行了数值研究,结果表明:修正cKdV方程既存在稳定的孤立波解,也存在不稳定的孤立波解.