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 development of science and technology, there are a lot of nonlinear problems in many disciplines. Some of the nonlinear problems are described by nonlinear differential equations in order to understand the physical meaning of these nonlinear differential equations in depth. To obtain the exact solution of the equation is the most important step. So far, due to the complexity of the nonlinear differential equation itself. There is no uniform method to find the exact solutions of these nonlinear differential equations. Through the unremitting efforts of mathematicians and physicists, a series of methods, such as backscattering, have been developed for finding exact solutions. The Darboux transform method includes Backlund transform method, bilinear method, Li Qun method, homogeneous balance method, auxiliary equation method and so on. The auxiliary equation method has been paid more and more attention because of its directness, simplicity and efficiency. In this paper, the problem of solving nonlinear evolution equation is studied and discussed with the aid of auxiliary equation method. In this paper, the gKdV-qRLW equation is solved by using the auxiliary equation with a single higher term and the auxiliary equation with two higher terms, respectively. Generalized symmetric regular long wave equations and g Zakharov equations and Klein-Gordon-Zakharov equations with arbitrary order. The F- / G- expansion method is generalized. The generalized F- / G- expansion method is used to solve the variable coefficient mKd V equation, variable coefficient KD V equation and Gross-Pitaevskii equation of cubic to quintic dimension. The exact solution of the equation is 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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