非线性系统的对称性及暗方程研究

发布时间：2018-01-02 06:16

本文关键词：非线性系统的对称性及暗方程研究　出处：《华东师范大学》2017年博士论文　论文类型：学位论文

【摘要】：本文基于符号计算软件Maple,利用对称性理论、相容Riccati方程展开法(CRE方法)、优化系统直接构造方法以及高阶对称延拓理论,研究了数学物理方程中若干非线性模型的相关问题,主要包含以下三方面的内容:若干非线性偏微分方程非局域对称及精确解的构造;优化系统的分类;暗方程的分类及递推算子的构造.具体章节安排如下:第一章绪论部分,介绍了本文研究内容的理论背景和发展现状,包括对称理论、优化系统、暗方程以及符号计算,并阐明了本论文的选题和主要工作.第二章,首先分别基于Darboux变换和Painleve截断展开法构造了 Kaup-Kupershmidt(KK)方程的非局域对称和留数对称,并通过引入辅助变量,成功地将非局域对称局域化,利用经典Lie群理论求解扩大系统的Lie点对称,并通过有限变换和相似约化给出了原系统孤立波与椭圆周期波等相互作用解;其次,利用相容Riccati方程展开法,求得了 AB系统的一些新的精确解.第三章,基于构造一维优化系统的直接算法,研究了 2 + 1维Wu-Zhang(WZ)方程Lie对称群的一维优化分类问题,构造了 WZ方程的优化系统,并通过相似约化得到了十五类1 + 1维完整且不等价的约化系统,不仅包括经典的Boussinesq方程和WZ方程的稳态形式,还包括一些Painleve可积的方程.第四章,基于Kupershmidt暗KdV方程的定义和分类,首先,在高阶微分形式对称存在的假设下,通过线性延拓得到了暗修正KdV(MKdV)方程的完整分类,得到了十二类不等价的暗MKdV系统.与暗KdV方程比较,每一类暗MKdV方程都含有自由参数,当自由参数固定时,其中九类暗MKdV方程正好是暗KdV方程作合适的Miura变换的结果,而另外三类不能通过调整自由参数的值,由暗KdV方程通过Miura变换求得.其次,对暗方程的定义进行了推广,通过线性非齐次延拓得到了含有更多自由参数的推广的暗MKdV方程的不等价分类.通过选取合适的自由参数,推广的暗MKdV方程恰好可以退化到所有齐次延拓的暗MKdV方程.另外,分别选取暗MKdV方程和推广的暗MKdV方程其中两类,推导了它们的递推算子.第五章,对全文工作进行了简要的总结和讨论,并对接下来将要研究的工作做了进一步展望.
[Abstract]:This paper is based on the symbolic computing software Maple, using symmetry theory, consistent Riccati equation expansion method, optimization system direct construction method and high-order symmetric continuation theory. In this paper, the related problems of some nonlinear models in mathematical physics equations are studied, including the following three aspects: the construction of nonlocal symmetry and exact solutions of some nonlinear partial differential equations; Optimizing the classification of the system; The classification of dark equations and the construction of recursive operators. The specific chapters are arranged as follows: the first chapter introduces the theoretical background and development of the content of this paper, including symmetry theory, optimization system. Dark equation and symbolic calculation, and clarify the topic of this paper and the main work. Chapter two. Firstly, the nonlocal symmetry and residue symmetry of Kaup-Kupershmidt KK) equation are constructed based on Darboux transform and Painleve truncation expansion method, respectively. By introducing auxiliary variables, the nonlocal symmetry is successfully localized, and the classical Lie group theory is used to solve the Lie point symmetry of the extended system. The interaction solutions of solitary wave and elliptic periodic wave of the original system are given by finite transformation and similarity reduction. Secondly, some new exact solutions of AB system are obtained by using consistent Riccati equation expansion method. Chapter 3, based on the direct algorithm of constructing one-dimensional optimization system. In this paper, the one-dimensional optimization classification problem of Lie symmetric group for 21 dimensional Wu-Zhang WZ) equation is studied, and the optimization system of WZ equation is constructed. Through similarity reduction, we obtain 15 classes of 11 dimensional complete and non-equivalent reduced systems, including not only the classical Boussinesq equation and the steady state form of WZ equation. It also includes some Painleve integrable equations. Chapter 4th, based on the definition and classification of Kupershmidt dark KdV equation, firstly, under the hypothesis of the existence of higher order differential form symmetry. The complete classification of dark modified KDV MKdV) equations is obtained by linear extension, and 12 classes of non-equivalent dark MKdV systems are obtained, which are compared with dark KdV equations. Each kind of dark MKdV equation contains free parameters. When the free parameter is fixed, nine kinds of dark MKdV equations are the result of the appropriate Miura transformation of the dark KdV equation. The other three types can not be obtained by adjusting the values of free parameters by Miura transformation. Secondly, the definition of dark equation is generalized. By linear inhomogeneous extension, we obtain the non-equivalent classification of the extended dark MKdV equation with more free parameters, and select appropriate free parameters. The generalized dark MKdV equation can degenerate to all homogeneous continuation dark MKdV equations. In addition, dark MKdV equation and generalized dark MKdV equation are selected respectively. Their recursive operators are derived in Chapter 5th, and the work of this paper is briefly summarized and discussed, and the future work to be studied is prospected.
【学位授予单位】：华东师范大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O175;O152.5

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