节点文献
用标准的和推广的tanh函数展开法求解非线性发展方程
【作者】 李军红；
【导师】 张顺利；
【作者基本信息】 西北大学 ， 应用数学， 2010， 硕士
【摘要】 随着非线性科学的不断发展,人们对非线性现象的研究越来越深入,研究的领域也越来越广泛,比如：非线性发展方程解的求法以及解的存在性、唯一性、稳定性等.其中对精确解的研究始终是一个重要课题,虽然现在已经形成了许多求解非线性发展方程的方法,但这些方法都仅适用于某些类型方程,对所有的非线性发展方程,没有普遍适用的方法.本文利用标准的和推广的tanh函数展开法对一些非线性发展方程进行求解,得到了大量的解,充分展示了tanh函数展开法的有效性和实用性.本文的内容和结构安排如下：第一章介绍非线性发展方程精确解的几种常见解法,并简单地回顾了精确解研究的发展概况.第二章用标准的tanh函数展开法对一些非线性发展方程进行求解,包括(3+1)维的KP方程、sine-Gordon方程、Higher-order KdV-like方程和Klein-Gordon-type方程.第三章利用推广的tanh函数展开法求解Boussinesq-type方程和breaking方程.最后,对本文的工作进行总结与展望,并且提出了一些自己有待研究的问题.
【Abstract】 With the development of nonlinear science, the study of nonlinear phenomena is becoming more and more clear, and the field of the study becomes very wide, such as the method to solve evolution equations and the method to discuss the existence, uniqueness, stability of exact solutions. The study of exact solutions is an important topic. There are a number of methods to solve evolution equations, but they are only applicable to certain types of equations. In fact, there is no method can be worked on all nonlinear evolution equations.This paper, we use a standard tanh-function method and the extended tanh-function method to solve some nonlinear evolution equations, and obtain a large number of exact solutions. It shows the effectiveness of tanh-function method.The structure of this paper is as follows:Chapter 1, we introduce several common methods to solve nonlinear evolution equations, and briefly review the present condition for the study of exact solutions.Chapter 2, we use the standard tanh-function method to solve several nonlinear evolution equations, which include (3+1)-dimensional KP, sine-Gordon, Higher-order KdV-like and Klein-Gordon-type equations.Chapter 3, exact solutions of Boussinesq-type equation and breaking equation are obtained by using extended tanh-function method.In the end, we summarize the work of this paper, and bring up a number of problems to be solved in the future.