辅助方程法及一些非线性发展方程（组）的精确解

发布时间：2018-01-07 17:20

本文关键词：辅助方程法及一些非线性发展方程（组）的精确解　出处：《河南科技大学》2015年硕士论文　论文类型：学位论文

【摘要】：随着科技的不断发展,在许多学科领域中存在着大量的非线性问题,其中一部分非线性问题是利用非线性微分方程来描述的。为了能深入地了解这些非线性微分方程的物理意义,获得方程的精确解就成为最为重要的一步。到目前为止,由于非线性微分方程的本身复杂性,还没有一个统一的方法来求得这些非线性微分方程的精确解。因此,非线性微分方程(组)的精确求解不论在理论上,还是在应用领域里仍是一个非常有研究价值的课题。经过数学家和物理学家不懈的努力,现已发展出一系列用于求精确解的方法,如反散射方法、Darboux变换方法、Backlund变换方法、双线性方法、李群方法、齐次平衡法、Dressing方法、辅助方程法等等。在这些求解方法中,辅助方程法由于直接、简洁、有效,而广受重视。本文主要借助于辅助方程法,对非线性发展方程求解问题进行了研究和探讨,主要研究:(1)分别利用具单个高次项的辅助方程和具两个高次项的辅助方程求解了gKdV-qRLW方程、gKawahara方程、广义对称正则长波方程以及g Zakharov方程组和具任意次Klein-Gordon-Zakharov方程组。(2)将F/G-展开法做了推广,利用推广的F/G-展开法求解了变系数mKd V方程、变系数Kd V方程和(3+1)维三次-五次Gross-Pitaevskii方程,得到了方程的精确解。
[Abstract]:With the continuous development of science and technology in many fields there are many nonlinear problems, the nonlinear part of the problem is the use of nonlinear differential equations. In order to deeply understand the physical meaning of these nonlinear differential equations, exact solutions of equations obtained has become the most important step. So far, because the nonlinear differential equations of the complexity, there is not a unified method to obtain the exact solutions of these nonlinear differential equations. Therefore, nonlinear differential equation (Group) the exact solution either in theory or in the application area, is a very valuable research topic. Through the unremitting efforts of mathematicians and physicists, now the development of a series of methods for seeking exact solutions, such as inverse scattering method, Darboux transformation method, Backlund transformation method, bilinear method, Li Qun method, homogeneous Balance method, Dressing method and auxiliary equation method and so on. In this method, the auxiliary equation method due to the direct, concise, effective, and wide attention. This paper is mainly based on the auxiliary equation method, the problem of solving nonlinear evolution equations are studied and discussed, the main research: (1) respectively by using the auxiliary equation with a single high-order and two high-order auxiliary equation gKdV-qRLW equation, gKawahara equation, G equation and the generalized symmetric regularized long wave equations with arbitrary Zakharov and Klein-Gordon-Zakharov equations. (2) the F/G- expansion method was extended by using the extended F/G- expansion method to solve the mKd V equation with variable coefficients. The variable coefficient Kd equation and V (3+1) three - five dimensional Gross-Pitaevskii equation, exact solutions of these equations are obtained.

【学位授予单位】：河南科技大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O175.29

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