Integrable Discretization And Bi-Hamiltonian Structure Of Soliton Equation

The properties of some significant soliton equations in nonlinear mathematical physic-s and the relationships between the equations are studied. The thesis can be mainly di-vided into four parts:using the algebraization of the difference operator, the integrable discretization of the continuous generalized nonlinear Schrodinger (GNLS) equation is investigated. Moreover, the integrability and all linear reductions of the obtained discrete equation are discussed; The bi-Hamiltonian structures of some multi-component soliton equations are constructed and validated. Then some other integrable properties of these equations may be derived; The reciprocal transformations of some important CH type equations are discovered and hence the relationships between different hierarchies are es-tablished; With the help of the symbolic computation software Mathematica, a software package for automatically testing the bi-Hamiltonian operators of the soliton equation is developed. The specific chapters are arranged as follows.In chapter 1, an introduction is devoted to the research background and the current situation of the integrable discretization, bi-Hamiltonian theory, reciprocal transformation and symbolic computation. The major results of this dissertation are also stated.In chapter 2, applying the algebraization of the difference operator to the Lax pair of continuous GNLS equation, the discrete GNLS equation is obtained. Then the integrable properties of the discrete GNLS equation, such as the recursion operator, the symmetry and the conservation law, are also considered. Based on the theory of circulant matrices, all the linear reductions of the discrete GNLS equation are studied. A classical discrete NLS equation is obtained by one of the reductions.In chapter 3, based on the zero-curvature equation, the bi-Hamiltonian operators of the multi-component Novikov equation, the multi-component Yajima-Oikawa (YO) hierarchy and a multi-component CH type equation are constructed and validated by the multi-vector approach. The obtained bi-Hamiltonian structures lead to other results:the recursion operator and the infinite nonlocal symmetries of the multi-component Novikov equation; the classical YO equation and its infinitely many conserved quantities; the dual hierarchies of the multi-component CH type equation, i.e., the multi-component AKNS hierarchy and the multi-component KN hierarchy.In chapter 4, the relation between the CH equation and the Olver-Rosenau-Qiao (ORQ) equation is established through two reciprocal transformations between the CH equation, ORQ equation and the first negative of the KdV hierarchy respectively. More-over, a hybrid CH type equation of the CH equation and ORQ equation is related to the first negative flow of the KdV hierarchy by a reciprocal transformation, which can reduce to the above two reciprocal transformations.In chapter 5, with the aid of symbolic computing platform Mathematica and the multi-vector method which is established to prove the bi-Hamiltonian operators, a soft-ware package’MvBiHamiltonian’is developed. The software package can automatically validate the antisymmetry, the Jacobi identity of a operator and the compatibility of two Hamiltonian operators. It is worth noting that the software package can successfully test all Hamiltonian pairs in the form of differential operator summarized by Wang.In chapter 6, the main work of this thesis is summarized and concluded. In addition, the orientation of further study is pointed out.