Researches On Related Nonlinear Ntegrable System And Darboux Transformations Of Soliton Equations

In this thesis, isospectral eigenvalue problems, which contain two and three potentials respectively, are established. Starting from isospectral problems, Tu's scheme is applied to generate a well-known generalized Burgers equation hierarchy, a class of new MKdV-NLS equation hierarchy and a family of discrete nonlinear evolution equations. They are shown to be Liouville integrable Hamiltonian systems. Moreover, MKdV-NLS hierarchy have Binary Hamiltonian structures. Then the nonlinearization procedure is applied to the eigenvalue problem of MKdV-NLS hierarchy. Under Bargmann constraint, it is shown that Lax pairs are nonlinearized to be two finite-dimensional Liouville completely integrable system. ConstuctingLoop algebra G leads to integrable couplings of the generalized Burgers hierarchy, integrable couplings of the MKdV-NLS hierarchy and a class of expandingintegrable model of Dirac hierarchy. Finally, by using Darboux transformation method, various of Darboux matrices are obtained. Darboux transformations of the mixed nonlinear schr dinger equation, WKI equation and a family of new discrete soliton equations are constructed. Furthermore, their exact solutions are derived.