几个辅助方程与非线性发展方程的精确解

发布时间：2018-06-08 02:48

本文选题：非线性发展方程 + 精确解　；参考：《内蒙古师范大学》2017年硕士论文

【摘要】：非线性发展方程(组)的精确求解问题是孤立子理论中的重要研究方向之一,它包括提出新的精确求解方法与利用已经建立的方法给出非线性发展方程新的精确解等两个方面.由于非线性发展方程(组)的复杂性,建立一个统一的精确求解方法看来十分困难.但是针对非线性发展方程的个性特征,探索并找出相应的较为系统的精确求解的新方法是可以实现的,一旦建立了一种新的求解方法就有可能获得所研究的非线性发展方程的新解或特殊解且它们恰好能解释新的物理现象.利用已经建立的方法求解非线性发展方程(组)而获取新解也是精确求解所采用的重要手段,且因方便、快捷、能够解释新发现的物理现象而受到物理学家、工程技术人员的关注.因此,非线性发展方程的精确求解问题的研究具有推动求解理论和求解方法的建立,为实际问题的解决和解释提供有效工具.本文继许多专家、学者研究工作的基础上,利用exp(-φ(ζ))-展开法、G'/G-展开法和广义辅助方程法研究一些非线性发展方程并尝试构造其精确解.本文主要由四部分组成,具体安排如下:第一章简要概述孤立子理论的研究意义、辅助方程方法的应用状况以及本文的主要研究工作.第二章简要介绍exp(-φ(ζ))-展开法,并将其应用到(2+1)维耗散长水波方程、广义的变系数KdV-mKdV方程以及变系数(2+1)维Broer-Kaup方程中,获得了新的奇异行波解.将本文结果与其他文献中用不同方法给出的精确解进行比较,得出它们只是本文所给出的解的特殊情形.第三章简要介绍G'/G-展开法,并以常系数Newell方程、变系数Novikov-Veselov方程以及离散复立方-五次Ginzburg-Landau方程为例,求解获得含有自由参数的通解.由此说明了 G'/G-展开法可以用于求解变系数方程、复方程和离散方程等各种类型的非线性发展方程的求解.G'/G-展开法求解过程简单、直接,而且由于自由参数的任意性,所得的精确解也更加丰富.第四章介绍和分析广义辅助方程法,通过构造出适当的解的形式去求解(2+1)维 Calogero-Bogoyavlenskii-Schiff 方程及变系数组合 KdV方程获得更丰富的精确类孤子解.
[Abstract]:The exact solution of nonlinear evolution equations (systems) is one of the important research directions in the soliton theory. It includes two aspects: a new exact solution method and a new exact solution of the nonlinear evolution equation by using the established method. Because of the complexity of nonlinear evolution equations, it is very difficult to establish a unified exact solution method. However, in view of the personality characteristics of nonlinear evolution equations, it is possible to explore and find out a new method to solve the nonlinear evolution equations accurately and systematically. Once a new method is established, it is possible to obtain new solutions or special solutions of the nonlinear evolution equations studied and they can explain the new physical phenomena. Using the established method to solve nonlinear evolution equations (systems) and obtaining new solutions is also an important means of exact solution. It is also a physicist who can explain the newly discovered physical phenomena because of its convenience and rapidity. The attention of engineers and technicians. Therefore, the research on the exact solution of nonlinear evolution equations has promoted the establishment of solving theory and method, and provided an effective tool for the solution and interpretation of practical problems. In this paper, based on the research work of many experts and scholars, some nonlinear evolution equations are studied and their exact solutions are constructed by using expan- 蠁 (味 ~ (-) -expansion method) and generalized auxiliary equation method. This paper is composed of four parts. The main contents are as follows: in chapter one, the research significance of soliton theory, the application of auxiliary equation method and the main research work of this paper are briefly summarized. In chapter 2, expan- 蠁 (味 ~ + -expansion method) is briefly introduced and applied to the dissipative long water wave equation, the generalized variable coefficient KdV-mKdV equation and the variable coefficient 21) -dimensional Broer-Kaup equation. A new singular traveling wave solution is obtained by applying the method to the dissipative long water wave equation in ~ (21) D, the generalized variable coefficient KdV-mKdV equation and the variable coefficient ~ (21) -dimensional Broer-Kaup equation. By comparing the results of this paper with the exact solutions given by different methods in other literatures, it is found that they are only the special cases of the solutions given in this paper. In chapter 3, we briefly introduce the Gon / G- expansion method, and take the Newell equation with constant coefficients, the Novikov-Veselov equation with variable coefficients and the Ginzburg-Landau equation of discrete complex cubic order as examples to obtain the general solution with free parameters. It is shown that the GG / G- expansion method can be used to solve all kinds of nonlinear evolution equations, such as variable coefficient equation, compound equation and discrete equation. The process of solving the nonlinear evolution equation is simple and direct, and because of the arbitrariness of free parameter, the method can be used to solve all kinds of nonlinear evolution equations, such as variable coefficient equation, compound equation and discrete equation. The exact solutions obtained are also more abundant. In chapter 4, the generalized auxiliary equation method is introduced and analyzed. By constructing an appropriate solution form to solve the Calogero-Bogoyavlenskii-Schiff equation and the variable coefficient combined KDV equation, the exact soliton-like solutions are obtained.
【学位授予单位】：内蒙古师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175.29

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