非线性晶格方程的辛映射及其精确解

发布时间：2018-03-20 01:16

本文选题：辛映射　切入点：Lie点对称　出处：《山东科技大学》2017年硕士论文　论文类型：学位论文

【摘要】：本论文主要研究:离散的微分-差分方程族的可积性及其在恰当Bargmann约束下的双非线性化,获得有限维完全可积的Hamilton系统和可积辛映射,最后运用Lie点对称方法求解方程族的精确解.第一章简要叙述了孤立子理论的起源、研究现状和应用背景,详细介绍了可积系统、可积耦合的概念.第二章主要介绍了本课题所涉及的一般理论及方法:两种意义下的可积性——Liouville可积和Lax可积,离散可积系的迹恒等式、离散等谱问题的屠格式、对称约束下的双非线性化和常用的两种对称求解方法——经典Lie群法和修正的CK直接方法.第三章主要研究一族离散可积方程族,并建立其Hamilton结构.分为两部分,第一部分:提出一个离散的2×2阶矩阵谱问题,根据驻定的离散零曲率方程,求解得到一族微分-差分方程族,并建立其Hamilton结构.第二部分:运用迹恒等式生成Liouville可积的Hamilton方程.第四章主要研究方程族的双非线性化及利用Lie点对称求方程族的精确解.分为两部分,第一部分:根据适当的Bargmann对称约束,对离散可积方程族的Lax对和伴随Lax对进行双非线性化,将空间部分和时间部分分别约化为一个有限维的完全可积系统和一个可积辛映射.第二部分:基于单参数变换群,根据其无穷小生成元,求其延拓向量场.通过将原方程代入延拓向量场,得到新的无穷小生成元,从而对方程组进行求解.
[Abstract]:In this paper, we study the integrability of discrete differential-difference equations and its double nonlinearization under proper Bargmann constraints, and obtain finite-dimensional completely integrable Hamilton systems and integrable symplectic mappings. In the first chapter, the origin, research status and application background of soliton theory are briefly described, and the integrable system is introduced in detail. The second chapter introduces the general theories and methods of this subject: the integrability of Liouville and Lax integrability, the trace identity of discrete integrable system, the slaughtering scheme of discrete isospectral problem. In the third chapter, we study a family of discrete integrable equations and establish its Hamilton structure, which consists of two parts: classical Lie group method and modified CK direct method. In the first part, a discrete 2 脳 2 order matrix spectrum problem is proposed. According to the stationary discrete zero curvature equation, a family of differential-difference equations is obtained. In the second part, the Liouville integrable Hamilton equation is generated by the trace identity. Chapter 4th mainly studies the double nonlinearization of the equation family and the exact solution of the equation family by using the Lie point symmetry. It is divided into two parts. In the first part, according to the appropriate Bargmann symmetry constraints, the Lax pair and the adjoint Lax pair of discrete integrable equations are doubly nonlinear. The space part and the time part are reduced to a finite-dimensional completely integrable system and a integrable symplectic map respectively. By substituting the original equation into the extended vector field, a new infinitesimal generator is obtained and the equations are solved.
【学位授予单位】：山东科技大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175

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