Nonlinear Intrgrable Systems And Integrable Expanding Models

Discrete and continuous integrable systems and its integrable expanding model are presented in this paper. In the first chapter, historical orgin and some researches of soliton theory together with its research meaning are presented. In chapter 2, using Tu scheme, a discrete matrix spectral problem is introduced and a hierarchy of discrete integrable systems is derived. Their Hamiltonian structures are esteblelished. A new integrable symplectic map and a family of finite- dimension completely integrable systems are obtained via binary nonlinearization of the spectral problem. Finally, the representation of solutions for the discrete integrable systems is given. Secondly, we formulated some discrete integrable systems and gave associated Hamilton structure by means of the enlarging isospectral problem. Thirdly, A family of integrable lattice equations with four potentials is constructed from a new discrete 3×3 matrix spectral problem. The Hamiltonian structures of the integrable lattice equations in the family are derived by applying the discrete trace identity. And the Liouville integrability of resulting discrete Hamiltonian equations is demonstrated. In chapter 3, the expanding integrable models are studied. Starting from an integrable coupling of a new 4×4 matrix spectral problem, an integrable system of the above hierarchy is obtained by use of a trace variational identity. The Liouville integrability for the hierarchy of the resulting Hamiltonian equation is proved.