Tanh函数展开法和CRE方法在非线性偏微分方程中的应用

发布时间：2018-01-19 18:20

本文关键词： Painleve测试 tanh函数展开法 CRE方法　出处：《宁波大学》2015年硕士论文　论文类型：学位论文

【摘要】：非线性方程是非线性科学的重要领域之一,而对于这些非线性方程的求解无疑成为非线性科学研究的关键所在,也是非线性研究的难点所在.通过众多科学家的努力,人们已经建立和发展了不少求解非线性方程的方法,如反散射变换方法、达布变换方法、双线性方法和多线性方法、Painleve截断展开方法、函数展开方法等等.在此基础上,楼森岳教授又提出了一种既简单又易懂的方法“CRE方法”.本文先介绍了几种研究非线性偏微分方程的可积性的方法,之后利用推广的Painleve截断展开法求得了Burgers方程的一个新的解.作为预备知识,本文又介绍了非线性方程的双曲函数展开法.通过分析KdV方程和耦合KdV方程组,得到了一些孤立波解.通过对一些基本知识的掌握,很自然的引进CRE方法.先对mKdV方程先进行Riccati函数展开,进而得到nKdV方程的相容性条件,通过对相容方程的求解,得到方程新的特殊结构的解.
[Abstract]:Nonlinear equation is one of the important fields of nonlinear science, and the solution of these nonlinear equations is undoubtedly the key of nonlinear science research. Through the efforts of many scientists, many methods for solving nonlinear equations have been established and developed, such as backscattering transform method and Darboux transform method. Bilinear method and multilinear method Painleve truncation expansion method, function expansion method and so on. Professor Lou Sanyue also proposed a simple and easy to understand method called CRE method. This paper first introduces several methods to study the integrability of nonlinear partial differential equations. Then a new solution of Burgers equation is obtained by using the extended Painleve truncation expansion method as a preparatory knowledge. This paper also introduces the hyperbolic function expansion method for nonlinear equations. By analyzing the KdV equation and coupled KdV equations, some solitary wave solutions are obtained. Naturally, the CRE method is introduced. Firstly, the mKdV equation is expanded by Riccati function, and then the compatibility condition of nKdV equation is obtained, and the compatibility equation is solved. The solution of the new special structure of the equation is obtained.
【学位授予单位】：宁波大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O175.29

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