AC=BD及其在非线性方程中的应用

【作者】 姜东梅；

【导师】 张鸿庆；

【作者基本信息】 大连理工大学 ， 应用数学， 2003， 硕士

【摘要】 本文研究的主要内容为：以构造性的变换和符号计算为工具，来研究非线性波和可积系统中的一些问题：精确解、Liouville可积的发展方程族及其N-Hamilton结构、约束流、Lax表示、r-矩阵、对合系统及对合解。 第二章考虑非线性偏微分方程的精确解的构造：首先给出了C-D对和C-D可积系统的基本理论，然后研究了它们具体的应用：在张鸿庆教授的AC=BD的模式下，利用闫振亚博士的Riccati方程展开法求解Modified Improved Boussinesq方程的精确解，得到了方程的26个解，包括新的孤波解和周期解，其中包含文献[65]中所给出的解。 第三章研究了Lax可积的新的方程族和Liouville可积的N-Hamilton结构方面的问题：利用屠格式计算一个等谱问题，得到了一族Lax可积的广义热传导方程族，并且构造了它的Liouville可积的N-Hamilton结构，给出了此发展方程族的Bargman约束流的r-矩阵，一个新的对合系统和解的对合表示。 第四章研究了Maxwell方程组用外微分形式和余微分形式表示的形式，了解了Maxwell方程组和电荷守恒定律可以互相推得，尤其是由电荷守恒定律可以推得Maxwell方程组，这对于未知的物理定律的发现有着深远的影响。更多还原

【Abstract】 The major contents in this paper include:with the aid of many types of constructive transformations and symbolic computation, some topics in nonlinear waves and integrable system are studied, including exact solutions,Liouville integrable hie-rarchy,and its N-Hamilton structure, constraint flow, Lax representation, r-matrix,in-volutive system and involutive solutions.Chapter 2 is devoted to investigating exact solutions of nonlinear wave equations: Firstly, the basic theories of C-D pair and C-D integrable system are presented. Secondly, we choose some examples to illustrate them .Based on Prof. Zhang hong-qing’ AC=BD theory, using one of Dr.Yan zhenya’ transformations based on one Riccati equation, twenty-six families of exact solutions of Modified Improved Bou-sinnesq equation are found, including new solitary solution and periodic solution, except solutions given by literature[65].Chapter 3 concentrates on new Lax integrable hierarchies of equations and Liouville integrable N-Hamilton structures. A spectral problem is studied by using the Tu’s scheme , its Lax integrable hierarchies of equations and Liouville integrable Hamilton structures are obtained , r-matrix, new involutive system and involutive solutions of Bargman constraint flow of this hierarchy are found.In chapter 4,Maxwell equations presented in the form of exterior and codiff-erential are studied , and Maxwell equations and charge conservation law can be derived by each other. In particular , the derivation of Maxwell equations from charge conservation law is very important for the discovery of unknown physical law.更多还原