Non-linear Evolution Equation And Integrable System

This paper discusses the formulation of the nonlinear evolution hierarchy as well as their expanding integrable systems.In the first chapter,historical origin and some researches of soliton theory together with its research meaning are presented.In the second chapter,firstly based on an old Lie algebra,a six dimensional Lie algebra is constructed through the linear combination,then the corresponding loop algebra is obtained.From the loop algebra a generalized isospectral problems are designed.Using Tu scheme,an expanding integrable model of(2+1)-dimensional JM hierarchy is worked out.Secondly,on the base of a multi-component loop algebra,multi-component(2+1)-dimensional JM hierarchy is worked out by use of(2+1)-dimensional zero-curvature equation.In the third chapter,firstly,a discrete matrix spectral problem is introduced,a hierarchy of nonlinear lattice equation is obtained by Tu scheme,then its Hamilton structure is presented by use of the extended trace identity. Secondly,we obtain its three integrable coupling systems with the help of two semi-direct sum Lie algebras.Thirdly,by using a new algebraic system G_M,a multi-component discrete integrable hierarchy is obtained,and an new extended algebraic system G_jM of G_M are presented,from which the integrable couplings system of the multi-component lattice hierarchy are obtained.In the fourth chapter,with the help of the cycled numbers,a higher-dimensional loop algebra is constructed.By employing loop algebra A₃^*,a generalized Toda hierarchy is obtained with possessing 2-Hamilton structure,which is also reduced to the well-known Toda hierarchy.