The Supersymmetric Tu Hierarchy And Its Nonlinearization Of Spectral Problem

It is very important to obtain new supersymmetric integrable Hamiltonian sys-tems and establish various connections among them. In this thesis, a novel integrablehierarchy of equations called the supersymmetric Tu hierarchy is presented and thenonlinearizations of the spectral problems is preformed. A new finite-dimensional su-persymmetric Hamiltonian system is obtained. The thesis consists of three chapters.In chapter one, a brief history of the supersymmetric soliton equations and thenonlinearizations of spectral problems are summarized.In chapter two, the linear spectral problem due to Tu is generalized to supersym-metric Lie algebra first. Then the hierarchy of supersymmetric Tu equations and theirzero curvature representation, as well as recursion operators, are derived using the Tuscheme. Finally, the bi-Hamiltonian structure of the hierarchy of supersymmetric Tuequations is constructed by use of the supertrace identity.Chapter three is devoted to the nonlinearizations of spectral problem for the hi-erarchy of supersymmetric Tu equations. Under a Neumann constraint, a new finite-dimensional Hamiltonian system is obtained. Its Lax representation and thus conversedintegrals of motion are given. In the unconstrained space, it is shown the Lax matrixsatisfies an r-matrix relation and the conversed integrals of motion are in involutionpairwise. In the constrained space, the Dirac bracket is given.