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Integrable Hamiltonian Systems And Their Algebraic Structure

Posted on:2009-05-09Degree:DoctorType:Dissertation
Country:ChinaCandidate:L LuoFull Text:PDF
GTID:1110360272958894Subject:Basic mathematics
Abstract/Summary:PDF Full Text Request
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.
Keywords/Search Tags:hierarchy of nonlinear equations, zero curvature equation, the Hamilton structure, the Liouville integrability, conservation laws, integrable coupling system, algebraic structure, the Darboux transformation, exact solution
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