孤子方程的可积离散和双哈密顿结构

发布时间：2018-03-19 12:14

本文选题：孤子方程　切入点：可积离散　出处：《华东师范大学》2016年博士论文　论文类型：学位论文

【摘要】：本文主要研究非线性数学物理中一些重要的孤子方程的性质及它们之间的相互关系.大致分为以下四方面内容：运用差分算子代数化将连续广义非线性薛定谔(GNLS)方程可积离散化,并研究所得离散方程的一些可积性质和线性约化；构造并验证若干多分量孤子方程的双哈密顿结构,进而推导方程的其他可积性质：发现一些重要的Camassa-Holm(CH)型方程的reciprocal变换,以此建立不同方程族之间的关系；利用符号计算平台Mathematica开发了验证孤子方程双哈密顿算子的自动推演程序包.具体的章节安排如下：第一章,介绍了与本文相关的可积离散、双哈密顿理论、reciprocal变换、符号计算的研究背景与发展现状,并且概述了本文的主要工作.第二章,利用差分算子代数化的方法将连续GNLS方程Lax对离散化、得到了半离散GNLS方程,并考虑了其可积性质如递推算子、对称和守恒律.通过循环矩阵理论研究了该广义半离散方程所有的线性约化,特别的经过其中一个约化得到了一个经典的半离散NLS方程.第三章,通过零曲率方程构造了多分量Novikov方程、多分量Yajima-Oikawa(YO)族以及一个多分量CH型方程的双哈密顿算子,并用multi-vector方法验证.由这些多分量方程的双哈密顿结构可推导以下结果：多分量Novikov方程的递推算子和无穷多个非局域对称；经典的YO方程及其无穷多守恒量；多分量CH型方程所在族的对偶族：即多分量AKNS族和多分量KN族.第四章,以CH方程、Olver-Rosenau-Qiao(ORQ)方程与KdV族负一流之间的两个reciprocal变换为桥梁,建立了CH方程和ORQ方程之间的联系.构造了一个混合CH型方程(混合CH方程与ORQ方程)与KdV族负一流之间的reciprocal变换,特别地此变换的约化可得将CH方程和ORQ方程联系到KdV族负一流的上述两个reciprocal变换.第五章,基于符号计算软件Mathematica和验证双哈密顿算子的multi-vector万法,编写了程序包MvBiHamiltonian该程序包能自动验证算子的反对称性、Jacobi恒等式和两个哈密顿算子之间的相容性等.值得注意的是,该程序包成功地验证了王总结的所有微分算子形式的双哈密顿算子.第六章,总结归纳了本文的主要工作,并阐明了接下来的研究方向.
[Abstract]:In this paper, we mainly study the properties of some important soliton equations in nonlinear mathematical physics and their mutual relations. They are roughly divided into the following four aspects: the continuous generalized nonlinear Schrodinger GNLSs is algebraic by means of difference operator algebra. Equation integrable discretization, Some integrable properties and linear reduction of the discrete equations are obtained, and the double Hamiltonian structures of some multicomponent soliton equations are constructed and verified, and other integrable properties of the equations are derived. The reciprocal transformations of some important Camassa-Holmach type equations are found. Using the symbolic computing platform Mathematica to develop the automatic derivation package for verifying the double Hamiltonian operators of soliton equations. The specific chapters are arranged as follows: in Chapter 1, integrable discretization related to this paper is introduced. The background and development of the double Hamiltonian theory of reciprocal transformation, symbolic computation and the main work of this paper are summarized. In chapter 2, the Lax equations of continuous GNLS equations are discretized by using the algebraic method of difference operators, and the semi-discrete GNLS equations are obtained. The integrable properties such as recursive operator, symmetry and conservation law are considered. The linear reduction of the generalized semi-discrete equation is studied by the cyclic matrix theory. In chapter 3, we construct a multicomponent Novikov equation, a multicomponent Yajima-Oikawa Yo) family and a double Hamiltonian operator for a multicomponent Ch type equation. The following results can be derived from the double Hamiltonian structure of these multicomponent equations: the recursive operators and infinite nonlocal symmetries of the multicomponent Novikov equations, the classical YO equations and their infinite conserved quantities; The dual families of the families in which the multicomponent Ch equations belong, namely, the multicomponent AKNS family and the multicomponent KN family. In chapter 4th, the two reciprocal transformations between the Ch equation and the negative first-class KdV family are used as a bridge. The relation between Ch equation and ORQ equation is established, and the reciprocal transformation between mixed Ch equation (mixed Ch equation and ORQ equation) and KdV family is constructed. In particular, the reduction of this transformation can be used to link Ch equation and ORQ equation to the KdV family negative first-class two reciprocal transformations. Chapter 5th, based on the symbolic computing software Mathematica and the multi-vector Wan method to verify the double Hamiltonian operator, A package MvBiHamiltonian is written, which can automatically verify the antisymmetry Jacobi identity of the operator and the compatibility between the two Hamiltonian operators. This package successfully verifies all the differential operators in the form of double Hamiltonian operators summarized by Wang. Chapter 6th summarizes the main work of this paper and illustrates the future research directions.
【学位授予单位】：华东师范大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O175

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