Investigations On Nonlinear Integrable Systems And Integrable Expansion

Integrable discrete and continuous systems and their integrable expandingmodels are presented in this paper. In chapter 2, firstly, a discrete matrix spectralproblem is introduced and a hierarchy of discrete integrable systems is derived.Their Hamiltonian structures are esteblelished, and it is shown that the resultingdiscrete systems are all Liouville integrable. A new integrable symplectic mapand a family of finite-dimension completely integrable systems are obtained viabinary nonlinearization of the spectral problem. Finally, the representation ofsolutions for the discrete integrable systems is given. Secondly, we formulatedsome discrete integrable systems and gave associated Hamilton structure bymeans of the enlarging isospectral problem. In chapter 3, at first, a highdismensional loop algebra (?)₁ is constructed and new isospectral problem isdevised according to the given loop algebra. As its applications, a newintegrable hierarchy is obtained which can be reduced to AKNS hierarchy andBPT hierarchy. Secondly, a matrix loop algebras (?)_3M is given, from which, anisospectral problem is formulated and a multi-component Liouville integrablehierarchy is presented by Tu-pattern, also, its Hamiltonian structure is given,which can be reduced to AKNS hierarchy.