The Generalizations Of Dirac Integrable Hierarchy And Bi-hamiltonian Structures

This thesis mainly studies the generalization of Dirac type integrable hierarchy and its bi-Hamiltonian structure.Looking for integrable system and integrable hierarchy is a hot topic in mathematical physics research,which has good theoretical significance and research value.The trace constant method(Tu scheme)proposed by Guizhang Tu is a simple and effective way to generate an integrable Hamiltonian structure.This thesis first briefly summarizes the generation and development process of soliton theory,gives some preparatory knowledge needed in this paper,and introduces the main method of this paper--Tu scheme.Then,based on the introduction of the trial function h under the Lie algebra sl(2,R)and so(3,R),the classical Dirac integrable hierarchy is generalized by the Tu scheme and its bi-Hamilton structure is obtained.At the same time,we get the generalization of super Dirac integrable hierarchy.Next,bi-integrable coupling of Dirac integrable hierarchy is studied based on matrix loop algebra.Finally,the complete theory of integrable couplingsystem is proposed for the first time.