两类非线性发展方程的有界行波解及其显式表达式

发布时间：2018-07-05 18:09

本文选题：有界行波解 + 双曲函数展开法　；参考：《贵州民族大学》2016年硕士论文

【摘要】：对非线性发展方程的研究是自然科学和工程技术中一个非常重要的课题,随着近年来研究的不断深入,许多作者已经得到了一些非线性发展方程的研究成果。本文利用平面动力系统理论、待定系数法、双曲函数展开法、指数函数展开法等方法对非线性发展方程的有界行波解的精确表达式进行研究,具体以下列方程为例:1.广义Kd V-Burgers-Kuramoto方程:2.Zakharov-Rubenchik方程:首先对方程(Ⅰ)作行波变换,再进行一次降幂运算,得出其等价的平面动力系统,根据雅克比行列式的特征值特点对其进行定性分析,利用齐次平衡法和双曲函数展开法给出了系统新的孤波解的精确表达式。对方程(II)先进行行波变换,再将方程化成与之对应的平面动力系统,利用平面动力系统理论和方法对其进行有限远处奇点分析,得出方程(II)存在一条同宿轨和一条异宿轨。根据等价平面动力系统理论的同宿轨和异宿轨与方程(II)的钟状孤波解和扭状孤波解之间的对应关系,利用待定系数法和Maple软件得到了方程(II)的钟状孤波解和扭状孤波解的显式表达式。
[Abstract]:The study of nonlinear evolution equations is a very important subject in natural science and engineering technology. With the development of research in recent years, many authors have obtained some research results of nonlinear evolution equations. In this paper, the exact expressions of the bounded traveling wave solutions of nonlinear evolution equations are studied by means of plane dynamic system theory, undetermined coefficient method, hyperbolic function expansion method and exponential function expansion method. The following equations are taken as an example: 1. Generalized Kd V-Burgers-Kuramoto equation: 2. Zakharov-Rubenchik equation: first of all, the equation (I) is transformed by traveling wave, and then the equivalent plane dynamic system is obtained. By using the homogeneous equilibrium method and the hyperbolic function expansion method, the exact expression of the new solitary wave solution of the system is given. The equation (II) is transformed by traveling wave first and then transformed into a plane dynamic system corresponding to it. The singular point of equation (II) is analyzed in finite distance by using the theory and method of plane dynamic system. It is concluded that there exists a homoclinic orbit and an heteroclinic orbit in equation (II). According to the correspondence between homoclinic orbit and heteroclinic orbit of equivalent plane dynamical system theory and bell solitary wave solution and torsional solitary wave solution of equation (II), The explicit expressions of bell solitary wave solution and torsional solitary wave solution of equation (II) are obtained by using undetermined coefficient method and Maple software.
【学位授予单位】：贵州民族大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O175.29

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