可积哈密顿系统及其代数结构

【作者】 罗琳；

【导师】 范恩贵；

【作者基本信息】 复旦大学 ， 基础数学， 2008， 博士

【摘要】 本文研究内容主要涉及可积系统的四个方面:与连续谱问题相联系的无穷维和有限维Hamilton系统;与离散谱问题相联系的Hamilton系统和无穷守恒律;可积耦合系统零曲率方程的代数结构;非线性方程族的Darboux变换和精确解.第一章,作为与本文相关的研究背景,简要综述了孤立子理论的产生和发展过程,特别针对性地介绍了近年来国内外在可积系统方面的研究成果和发展状况.第二章,从一个广义Kaup-Newell谱问题出发,导出广义Kaup-Newell方程族,并利用迹恒等式建立该方程族的双Hamilton结构.借助非线性化方法,将广义Kaup-Newell谱问题非线性化为有限维完全可积的Hamilton系统,进而利用可换流的对合解给出孤子方程族解的对合表示.利用李代数的半直和思想构造了广义Kaup-Newell方程族的可积耦合系统,并得到了耦合系统的拟Hamilton结构.第三章,从一个离散谱问题出发,导出离散的正负发展方程族,利用离散形式的迹恒等式建立了这两个方程族的双Hamilton结构,并进一步获得了相应方程族的无穷守恒律.第四章,基于一种特殊的李代数半直和,通过定义Lax算子的交换关系,研究了连续和离散耦合系统的零曲率表示的代数结构,并将这种结构分别应用到AKNS方程族和Volterra离散族所生成的等谱族的τ—对称代数.第五章,从另一个广义Kaup-Newell谱问题出发,导出了相应的非线性发展方程族,并进一步借助Lax对的规范变换,统一地构造了整个方程族的Darboux变换和精确解.更多还原

【Abstract】 The focus of this dissertation is on the following four topics in the theory of integrable system:infinite-dimensional and finite-dimensional Hamiltonian systems related to a continuous spectral problem;the Hamiltonian systems and infinite conservation laws associated with a discrete spectral problem;the algebraic structure of the zero curvature for an integrable coupling system;the Darboux transformation and exact solutions for a hierarchy of nonlinear equations.Chapter 1 is devoted to the research background in connection with the dissertation. We briefly outline the origination and development of the soliton theory. Subsequently,we summarize the recent development and achievement in the integrable systems at home and abroad.In chapter 2,starting from a generalized Kaup-Newell(KN) spectral problem, we derive a generalized KN hierarchy.It is shown that the hierarchy is integrable in Liouville sense and possesses bi-Hamiltonian structure via the trace identity. Moreover,the spectral problem can be nonlineared into a finite-dimensional completely integrable Hamiltonian system through the nonlinearization of the Lax pair. The involutive representation of the solutions for the corresponding soliton equations is given due to the involutive solutions of the commuting flows.Finally,we construct the integrable coupling system of the generalized KN hierarchy and its quasi-Hamiltonian structure by using the conception of semi-direct sums of Lie algebraic.In chapter 3,by making use of a concrete spectral problem,we deduce a positive and a negative evolution equation hierarchies.It is found that the both hierarchies possess bi-Hamiltonian structure.The infinite conservation laws of the two hierarchies are further obtained based on their Lax pairs.In chapter 4,based on a kind of special semi-direct sums of Lie algebra,we focus on the algebraic structure of zero curvature representations associated with continuous and discrete integrable couplings(including continue and concrete cases) by defining the commutator of Lax operator.Further we apply such Lie structures in the AKNS and the Velterra integrable couplings to generateτ-symmetry algebras of the corresponding isospectral flows.In chapter 5,starting from another generalized KN spectral problem,we present a generalized KN hierarchy of nonlinear evolution equations.Following the idea of gauge transformation of Lax pairs,we further provide an uniformly the Darboux transformation and corresponding exact solutions for the whole hierarchy.更多还原