Researches On The Integrable Systems Of The Soliton Equations

Two topics of the dissertation research about the integrable system of the soliton equations include: the formulation and integrability 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, an integrable system possessing bi-Hamiltonian structure is presented by choosing V with spectral potentials by use of the zero-curvature equation U_t—V_x+[U,V]=0 andTu scheme, which reduces to a system similar to the AKNS hierarchy and heat conduction equation; Secondly, a type of (2+1)-dimensional multi-component integrable hierarchy is obtained with the help of a (2+1)-dimensional zero-curvature equation and Tu scheme. As reduction cases, the (2+1)-dimensional multi-component AKNS hierarchy and KN hierarchy are worked out respectively; atlast, a loop algebra (X|~) with 3M dimensions is constructed by using of someproperties of exterior algebra, which is devoted to establishing many isospectral problems. As its application, a multi-component system similar to the BPT hierarchy is obtained. In the third chapter, firstly, new expanding loop algebras of the loop algebras presented in the second chapter are constructed, then three kinds of corresponding expanding integrable models are engendered by employing proper isospectral problems and Tu sheme. Secondly, a 6-dimensional Lie algebra is established with the help of the linear combination of a subalgebra, then the extended integrable model of the muti-component KN hierarchy is presented by use of the generalized isospectral problem and Tu sheme. The simple approach to generate expanding integrable models can be used generally.