【作者】 孟莹；

【导师】 梅建琴；

【作者基本信息】 大连理工大学 ， 应用数学， 2010， 硕士

【摘要】 非线性演化方程是描述物理现象的一类重要数学模型,也是非线性物理特别是孤立子理论最前沿的研究课题之一.非线性演化方程精确解和可积性的研究有助于弄清物质在非线性作用下的运动规律,对相应物理现象的科学解释和工程应用起到重要作用.在非线性演化方程的研究中,寻找方程的行波解、构造多孤子解、Painleve可积性质的检验等经常遇到复杂的符号计算和推理,有的是人力难以完成的,因此妨碍了这些问题的深入剖析.近年来,符号计算的蓬勃发展,极大地推动了非线性演化方程的研究,成果不断涌现,尤其是新的求解方法层出不穷.本文以非线性演化方程为研究对象,借助于符号计算,主要研究了一类Schrodinger方程的Lax对,并对该类方程做了Painleve方程检验,求得了Boussinesq-Burgers的两类N重Darboux变换和几个精确解.主要工作如下：第一章绪论部分介绍了孤立子理论产生的背景、发展及研究方法,并对Schrodinger方程的研究做了简单的综述.第二章是"AC=BD"理论,介绍了该理论的基本思想及应用,并用"AC=BD"来描述其他常见的研究孤子方程的方法.第三章介绍了两类可积,一是Lax可积及求Lax对的方法—延拓结构法.二是Painleve可积及Painleve检验的方法和与之相关的Backlund变换Darboux变换.第四章研究了一类变系数Schrodinger方程,利用数学机械化先判断方程Painleve可积性,又讨论了方程的Lax可积性.第五章构造了Boussinesq-Burgers方程的Douboux变换,并在此基础上给出了该方程的两个新解.最后给出了本文的简短总结.更多还原

【Abstract】 Nonlinear evolution equation is an important mathematical model for describing physical phenomenon and an important field in the contempary study of nonlinear physics, especially in the study of soliton theory. The research on the explicit solution and integrability are helpful in clarifying the movement of matter under the nonlinear interactivities and plays an important role in scientifically explaining of the corresponding physical phenomenon and engineering ap-plication. Many research topics, such as searching for exact explicit solutions, multi-soliton solution, the Painleve test etc., often involve a large amount of tedious algebra auxiliary reasoning or calculations which can become unmanageable in practice, In recent years, the development of symbolic computation accelerates the research of nonlinear evolution equation greatly. Many new methods for constructing exact solutions of nonlinear evolution equations are proposed.This dissertation equations with the aid of symbolic computation, mainly studies Lax pairs and Painleve test of a kind of Schrodinger equation. And as for Boussinesq-Burgers equation, two types of N-fold Darboux transformations and some solutions are derived. The article consists of the following parts:Chapter 1 introduces the backgound, development and research method of soliton theory and gives Schrodinger equations a summery preparing for Chapter 4.Chapter 2 states the theory of "AC=BD" including babsic thought and its application describing other common research method.In Chapter 3, two kinds of integrability are introduced. One is Lax integrability including prolongation structure method for deriving Lax pair of equations. The other is Painleve inte-grability which contains Painleve test method, Backlund transformation and Darboux transfor-mation.Chapter 4 studies a kind of Schrodinger equation. Making use of symbolic computation we study Painleve integrability first, and then Lax integrability is discussed.Chapter 5 constructs the Darboux transformation of Boussinesq-Burgers equation. And on that base some exact solutions are presented.At last a short summery of the dissertation is given.更多还原