Darboux变换在孤子方程中的应用

发布时间：2018-09-10 10:16

【摘要】：Darboux变换是一种研究孤子方程精确解较为直接和有效的方法。它建立了同一个方程的两个不同解之间的联系,从而可以从方程的一个平凡解得到许多非平凡解。本文主要研究了两个与3×3矩阵谱问题相联系的孤子方程的Darboux变换及其应用。首先,介绍了孤立子及Darboux变换的发展背景和基本理论。其次,通过适当引入谱问题的规范变换,构造出了广义TD方程的一阶及N阶Darboux变换。选取平凡的初始解,应用一阶Darboux变换,得到广义TD方程的两个非平凡的精确解。最后,讨论了一个微分-积分方程的Darboux变换及其精确解。
[Abstract]:Darboux transform is a direct and effective method to study the exact solutions of soliton equations. It establishes the relation between two different solutions of the same equation, and thus obtains many nontrivial solutions from one trivial solution of the equation. In this paper, we study the Darboux transformation of two soliton equations associated with 3 脳 3 matrix spectral problems and their applications. Firstly, the background and basic theory of soliton and Darboux transform are introduced. Secondly, the first order and N order Darboux transformation of the generalized TD equation is constructed by introducing the normal transformation of the spectral problem. Two nontrivial exact solutions of the generalized TD equation are obtained by using the first order Darboux transformation. Finally, the Darboux transformation of a differential-integral equation and its exact solution are discussed.
【学位授予单位】：华侨大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175

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