一般Hoof族研究

发布时间：2018-03-21 17:15

本文选题：Hopf族　切入点：对称　出处：《华东师范大学》2016年硕士论文　论文类型：学位论文

【摘要】：本文主要利用对称理论,以计算机代数系统软件Maple为工具,研究一般Hopf方程和Hopf族的可积性质,并对方程进一步求解。第一章绪论部分,主要介绍了孤立子理论的发展和现状,以及对称理论、符号计算在孤子理论中的重要作用。第二章给出了对称群、守恒律和群不变解等预备知识,为后面的研究做铺垫。第三章从定义出发,研究了Hopf方程的对称、强对称,得到了高阶对称的形式。接着给出了遗传算子的计算步骤,在遗传算子的作用下,提出了Hopf输运(扩散)方程,并研究其可积性质和具体方程的求解。第四章研究了一般Hopf方程和Hopf族。一般Hopf方程存在无穷多对称,可以用含有任意维任意函数的对称表示。本文提出了新的不变函数定义,通过研究不变函数、不变算子、强对称、遗传算子,利用一般对称理论,推导出了拥有无穷多对称和Lax对的Hopf族,并通过hodograph变换得出了Hopf方程的一般解。对于二阶Hopf方程(Hopf扩散方程,也称为Hopf输运方程),本文给出了五类无穷多对称。特别地,Hopf输运方程存在主枝解,这个解与通常的一般线性扩散方程的解相联系。本章最后给出了扩散方程几类特殊的精确群不变解。第五章对整篇文章的成果进行了总结和讨论。
[Abstract]:In this paper, the integrable properties of the general Hopf equation and the Hopf family are studied by using the symmetric theory and the computer algebraic system software Maple, and the equations are solved further. In the first chapter, the introduction mainly introduces the development and present situation of the soliton theory. In the second chapter, we give the basic knowledge of symmetric group, conservation law and group invariant solution. In chapter 3, we study the symmetry of Hopf equation from the definition of symmetry group, conservation law and group invariant solution. The form of high order symmetry is obtained by strong symmetry. Then, the calculation steps of genetic operator are given. Under the action of genetic operator, the Hopf transport (diffusion) equation is proposed. In chapter 4, the general Hopf equation and Hopf family are studied. There are infinite symmetries in the general Hopf equation, which can be expressed by the symmetry of arbitrary function with arbitrary dimension. In this paper, a new definition of invariant function is proposed. By studying invariant function, invariant operator, strong symmetry, genetic operator, and using the general symmetry theory, the Hopf family with infinite symmetry and Lax pairs is derived. The general solution of the Hopf equation is obtained by hodograph transformation. For the second order Hopf equation, also known as the Hopf transport equation, five classes of infinite symmetries are given. This solution is related to the solution of the general linear diffusion equation. In the end of this chapter, some special exact group invariant solutions of the diffusion equation are given. In chapter 5, the results of the whole paper are summarized and discussed.
【学位授予单位】：华东师范大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O175

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