The Positive, Negative Soliton Equations Of Dirac Hierarchy And The Finite-dimension Integrable Systems

In this paper, we discuss the negative soliton equations and the mixed soliton equa-tions, based on the research of the positive soliton equations of Dirac spectral problem.Then we obtain the integrable systems in the framework of Lie-Poisson structure from dealing with the positive, the negative and the mixed spectral problems with the nonlineariza-tion approach of eigenvalue problems, we also discuss the integrability of those finite-dimensional integrable systems. In addition,we straighten the corresponding flows in the framework of Hamilton-Jacobi theory.