The Integrable Systems With The Hamilton Form And Expanding Integrable Models

It has been an important subject to find a new integrable system and its expanding in thetheory of integrable systems. The main idea of this paper is to formulate a series of newintegrable equations in the Hamilton form which is based on Tu Scheme, and figure out theexpanding integrable models of these hierarchies, which is integrable coupling. Firstly, basedon a new Loop algebra and an isospectral problem designed, an integrable hierarchy ofequations which is reduced the NLS-MKDV hierarchy of equations are worked by using Tuscheme. Lastly, Hamilton structure and integrable coupling are obtained; Secondly, startingfrom a Loop algebraA%2, a new isospectral problem is designed. Moreover, an integrablehierarchy of equations with Hamilton structure are worked. Besides, high dimension Loopalgebra from the expanding ofA%2is applied to obtain the integrable coupling; Lastly, a newdiscrete isospectral problem is designed to deduce a discrete integrable systems with Hamiltonstructure and its expanding integrable models.