摘  要:本文利用辅助函数方法对广义反应Duffing方程进行了研究,获得了一些该方程的周期波解，孤立波解和纽子波解.

本文分为三部分.第一部分介绍了非线性波动方程的研究现状及意义.第二部分介绍了辅助函数方法，利用该方法求广义Duffing方程的精确解.第三部分简要说明了论文的研究成果.

关键词：广义反应Duffing方程;辅助函数方法;Jacobi椭圆函数;精确解;行波

Abstract:In this paper, a general nonlinear Duffing equation is studied by the auxiliary function method, some periodic wave, solitary wave and kink wave solutions, which expressed various functions are obtained. 

This paper consists of three parts. The present researching situations and significances of Nonlinear wave equations are introduced in the first part. In the second part, the auxiliary function method is introduced and some exact solutions of the generalized nonlinear Duffing equations are obtained by using the method. The research results of the paper are analyzed in the third part.

Key words: Duffing equation; Auxiliary function method; Jacobi elliptic function; Exact solution; Travelling wave

目录

第一章 绪  论-1

1.1非线性波动方程的研究意义及现状-1

1.2本文的主要研究内容-1

第二章 广义反应Duffing方程的精确解-3

2.1 辅助函数法介绍-3

2.2 Duffing方程的精确解-4

2.2.1当满足方程（5）时，Duffing模型的精确行波解-4

2.2.2当满足方程（6）时，Duffing模型的精确行波解-12

2.2.3当满足方程（7）时，Duffing模型的精确行波解-38

第三章 结  论-44

参考文献-45

附  录 -47

致  谢-49