2+1维孤子方程的Darboux变换及其精确解
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摘要
本文以数学机械化思想为指导,在导师张鸿庆教授提出的AC=BD理论下,借助于符号计算软件Maple,来研究非线性微分方程求解中的一些问题。其中Darboux变换就是一种十分有效的方法,它从孤子方程的一个平凡解出发能够求出一系列精确解。文章主要内容可概括如下:
第一章介绍了数学机械化及其相关学科的发展,围绕微分方程的求解理论和计算机代数的关系,简述了相关方面的国内外研究和发展概况。
第二章考虑了微分方程的AC=BD理论,介绍了C-D对理论的基本内容和思想,AC=BD理论侧重于对微分方程变换的机械化构造,可以把复杂问题转化为简单问题,把变系数问题转化为常系数问题,把非线性问题转化为线性问题来解决。
第三章介绍了Darboux变换研究的历史和它的一般理论。
四五两章以Levi,KPII和mKP方程为例,借助谱问题的规范变换,构造出这些2+1维孤子方程的Darboux变换,并且利用Darboux变换求得这些2+1维孤子方程的新孤子解。
In this dissertation,under the guidance of mathematical mechanization and the AC=BD theory put forward by Prof.Zhang Hongqing.and by means of symbolic computation software Maple,some topics in symbolic integration and differential equation are studied. Among the various approaches,the Darboux transformation is a very powerful tool,which can be used to find explicit solutions of soliton equations from a trivial seed.
Chapter 1 is to introduce the related development of mathematical physics mechanization, emphasizing on the relation between differential equations and computer algebra.We give an introduction of mathematical physics mechanization at home and abroad in summary.
Chapter 2 concerns the construction of transformation of differential equations under the uniform frame work of AC=BD model theory introduced by Prof.H.Q.Zhang.The basic theory of C-D pair is presented.
Chapter 3 is to introduce the history and theory of Darboux transformation.
In chapter 4 and 5,we take Levi KPII and mKP soliton equations for example,based on the gauge transformation of the spectral problem,we obtain the Darboux transformations of some(2+1) dimensional soliton equations and use the Darboux transformation for generating the exact solutions of the(2+1) dimensional soliton equations.
第一章介绍了数学机械化及其相关学科的发展,围绕微分方程的求解理论和计算机代数的关系,简述了相关方面的国内外研究和发展概况。
第二章考虑了微分方程的AC=BD理论,介绍了C-D对理论的基本内容和思想,AC=BD理论侧重于对微分方程变换的机械化构造,可以把复杂问题转化为简单问题,把变系数问题转化为常系数问题,把非线性问题转化为线性问题来解决。
第三章介绍了Darboux变换研究的历史和它的一般理论。
四五两章以Levi,KPII和mKP方程为例,借助谱问题的规范变换,构造出这些2+1维孤子方程的Darboux变换,并且利用Darboux变换求得这些2+1维孤子方程的新孤子解。
In this dissertation,under the guidance of mathematical mechanization and the AC=BD theory put forward by Prof.Zhang Hongqing.and by means of symbolic computation software Maple,some topics in symbolic integration and differential equation are studied. Among the various approaches,the Darboux transformation is a very powerful tool,which can be used to find explicit solutions of soliton equations from a trivial seed.
Chapter 1 is to introduce the related development of mathematical physics mechanization, emphasizing on the relation between differential equations and computer algebra.We give an introduction of mathematical physics mechanization at home and abroad in summary.
Chapter 2 concerns the construction of transformation of differential equations under the uniform frame work of AC=BD model theory introduced by Prof.H.Q.Zhang.The basic theory of C-D pair is presented.
Chapter 3 is to introduce the history and theory of Darboux transformation.
In chapter 4 and 5,we take Levi KPII and mKP soliton equations for example,based on the gauge transformation of the spectral problem,we obtain the Darboux transformations of some(2+1) dimensional soliton equations and use the Darboux transformation for generating the exact solutions of the(2+1) dimensional soliton equations.
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