几个非线性发展方程的精确解

发布时间：2018-02-27 03:07

本文关键词： 非线性发展方程精确解孤立波解扩展的Jacobi椭圆函数展开法　出处：《江苏大学》2009年硕士论文　论文类型：学位论文

【摘要】： 本文主要围绕精确求解非线性发展方程(NEE)的若干问题进行了研究和探讨,重点对扩展的Jacobi椭圆函数展开法进行了改进,丰富和发展了已有的方法,求出了几类非线性发展方程的新的精确解,并研究了解的结构。本文章节和内容安排如下: 第一章介绍了非线性发展方程相关理论研究的历史和发展现状,并介绍了本文的主要工作。第二章介绍了一些和本文相关的基本概念、符号,给出了孤立子的概念及分类,同时对容易混淆的精确解、近似解和相似解做了说明。第三章介绍了几种精确求解非线性发展方程的重要方法。如传统的Jacobi椭圆函数展开法、扩展的Jacobi椭圆函数展开法、齐次平衡法、吴代数消元法。第四章对传统的Jacobi椭圆函数展开法进行了推广和改进,给出了多种扩展的Jacobi椭圆函数法中形式解的统一形式,并应用该方法研究了Klein-Gordon方程、mKdV-ZK方程和Zakharov方程的精确解。除了得到已有的大量结果外,还得到了许多有意义的新解。这一方法与传统的方法相比,具有形式统一、使用方便、得到的结果更全面等优点。这对于发现新的孤立子解,研究孤子的结构有着积极的意义。
[Abstract]:In this paper, some problems about the exact solution of nonlinear evolution equations are studied and discussed, and the extended Jacobi elliptic function expansion method is improved, which enriches and develops the existing methods. The new exact solutions of some nonlinear evolution equations are obtained, and the structure of the solution is studied. The chapters and contents of this paper are arranged as follows:. The first chapter introduces the history and development of nonlinear evolution equation theory, and introduces the main work of this paper. In the second chapter, we introduce some basic concepts, symbols, and the concept and classification of solitons. At the same time, we explain the exact solutions, approximate solutions and similar solutions which are easy to be confused. In chapter 3, several important methods for solving nonlinear evolution equations are introduced, such as the traditional Jacobi elliptic function expansion method, the extended Jacobi elliptic function expansion method, the homogeneous equilibrium method and the Wu algebraic elimination method. In Chapter 4th, the traditional Jacobi elliptic function expansion method is generalized and improved, and the unified form of formal solutions in various extended Jacobi elliptic function methods is given. The exact solutions of mKdV-ZK equation and Zakharov equation of Klein-Gordon equation are studied by using this method. In addition to a large number of available results, many meaningful new solutions are obtained. Compared with the traditional method, this method has the advantages of uniform form and convenient use. The results obtained are more comprehensive and so on, which has a positive significance for finding new soliton solutions and studying the structure of solitons.
【学位授予单位】：江苏大学
【学位级别】：硕士
【学位授予年份】：2009
【分类号】：O175.29

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