几类非线性微分方程的对称分析

发布时间：2018-07-11 22:03

本文选题：对称约化 + 破裂孤子方程　；参考：《北方民族大学》2017年硕士论文

【摘要】：本文主要研究了几类非线性微分方程的对称分析。基于符号计算软件Maple,本文利用楼直接方法研究了一个(2+1)-维Toda-like晶格方程的对称变换。并基于求得的对称变换,得到了这个微分差分方程一个新的类孤子解,并给出数值算例。同时利用经典李群法对(2+1)-维非等谱破裂孤子方程组以及它的Lax对进行对称约化,并假设谱参数为一个额外的场,得到新的约化方程和Lax对,通过比较约化后的方程和约化后的Lax对的相容条件,我们发现约化的Lax对恰好是约化方程的Lax对,并通过一个约化后的Lax对求得了原方程的精确解。本论文的结构如下:引言部分介绍了可积系统和孤子理论的历史背景和发展状况,并且阐述了非线性微分方程的求解方法和对称分析,同时介绍了可积系统的理论背景及其广泛应用。第二、三章是文章的主要内容。最后对整篇文章进行总结。
[Abstract]:In this paper, the symmetry analysis of several nonlinear differential equations is studied. Based on the symbolic computing software Maple. the symmetric transformation of a (21) -dimensional Toda-like lattice equation is studied by building direct method. Based on the obtained symmetry transformation, a new soliton-like solution of the differential difference equation is obtained, and a numerical example is given. At the same time, the classical Li Qun method is used to symmetrically reduce (21) -dimensional nonisospectral rupture soliton equations and its lax pairs. The spectral parameters are assumed to be an extra field, and a new reductive equation and lax pair are obtained. By comparing the compatibility conditions between the reduced equation and the reduced lax pair, we find that the reduced lax pair is exactly the lax pair of the reduced equation, and the exact solution of the original equation is obtained by a reduced lax pair. The structure of this paper is as follows: the introduction introduces the historical background and development of the integrable system and soliton theory, and expounds the methods of solving nonlinear differential equations and symmetry analysis. At the same time, the theoretical background of integrable system and its wide application are introduced. The second, third chapter is the main content of the article. Finally, the article is summarized.
【学位授予单位】：北方民族大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175

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