【作者】 李玮；

【导师】 张鸿庆；

【作者基本信息】 大连理工大学 ， 应用数学， 2013， 博士

【摘要】 非线性偏微分方程是数学物理等诸多研究领域备受关注的研究对象.本文根据数学机械化思想,AC=BD模式,活动标架理论,围绕非线性偏微分方程这一课题展开了三个方面的研究：非线性偏微分方程组的精确求解,微分方程(组)的微分不变量完备系统的求解,等变活动标架法求解部分非线性偏微分方程.本文由以下六章组成：第一章对本文涉及的学科和理论：孤立子理论,数学机械化,活动标架理论,微分不变量理论,数学物理方程的精确求解及这些领域的发展情况和研究成果进行概述.最后,简要介绍本论文的选题及主要工作;第二章主要介绍与本论文相关的预备知识,即李群及其在微分方程中应用的相关基础知识;第三章简要概述AC=BD理论,介绍该理论在微分方程求解方面的应用；第四章在第三章理论的指导下,介绍求解非线性偏微分方程组精确解的一种新方法,即扩展的子方程展开法.本章以广义Schrodinger-Boussinesq方程和耦合非线性Klein-Gordon-Schrodinger方程为例,证明了该方法的有效性；第五章介绍等变活动标架理论,并以其为基础研究了该理论在微分方程(组)的微分不变量完备系统求解中的应用.求解了数学物理方程：Hirota-Ramani方程和Drinfel’d-Sokolov-Wilson方程的微分不变量完备系统；第六章结合第五章理论结果,探索等变活动标架理论在非线性偏微分方程求解方面的应用,给出求解部分非线性偏微分方程的一种新方法,即活动标架等变法,进一步丰富了AC=BD模式.更多还原

【Abstract】 Nonlinear partial differential equations are important mathematical models in various fields of sciences such as mathematics and physics. In this dissertation, based on mathematical mecha-nization idea, AC=BD model and moving frames theory, some topics are studied including exact solutions of nonlinear partial differential equations, complete system of differential invariants of differential equations and the method of equivariant moving frames to solve partial differential equations. There are six chapters in this dissertation:In Chapter1, the development and research achievements of soliton theory, mathematical mechanization, moving frames theory, differential invariant theory, exact solutions of mathe-matical physics equations and the brief of research work in this dissertation are introduced.In Chapter2, the basic knowledge associated with this dissertation is introduced, including some fundamental knowledge about Lie groups and it’s applications to differential equations.In Chapter3, AC=BD theory and its applications to differential equations are introduced.In Chapter4, based on the theory in Chapter3, the extended coupled sub-equation ex-pansion method is improved to solve nonlinear partial differential equations. The generalized Schrodinger-Boussinesq equation and coupled nonlinear Klein-Gordon-Schrodinger equations are given to prove the validity and advantages of this method.In Chapter5, the moving frames theory and it’s application to find the complete system of differential invariants of differential equations is introduced. The constructive computational al-gorithms to determine the complete systems of differential invariants of the mathematical physics equations:Hirota-Ramani equation and Drinfel’d-Sokolov-Wilson system are presented.In Chapter6, with the theoretical results in Chapter5, the application of moving frames theory to solve nonlinear partial differential equations is explored. A new method, the equivariant method of moving frames, to solve some nonlinear partial differential equations is given and enriches AC=BD model.更多还原