Researches On The Integrable Systems Of The Soliton Equations And The Expanding Integrable Model

The major contents in this paper include: the formulation of integrablehierarchies and the expanding integrable models. In the first chapter,historical origin and some researches of soliton theory together with itsresearch meaning are presented. In the second chapter, firstly,by using Tuscheme and the generalized zero curvature equation, a generalized(2+1)-dimensional Levi hierarchy of equations are obtained, respectively, viaconstruction of a loop algebra. Secondly, a new 3M dimensional loop algebra(?)_M with a simple commutative operation just like that in the loop algebra(?)₁ is constructed, which is devoted to establishing a new isospectral problem.As its application, a multi-component Guo hierarchy is worked out. Then withthe help of the extended trace identity, its Hamiltonian is obtained. Theexpanding integrable models are studied in the third chapter, firstly basedon the extended loop algebras of loop algebras in the second chapter, someexpanding integrable models are obtained. As its application, the expandingintegrable model of generalized (2+1)-dimensional Levi hierarchy of equationsis obtained. Secondly, a Lie algebra with 6 dimensions based on the linecombination of the Lie algebras A₁₁ is constructed, then its correspondingloop algebras is obtained and proposed a direct method for the expandingintegrable models. As their applications, the expanding integrable models ofthe multi-component Levi hierarchy is obtained. Last by constructing anexpanding loop algebra (?)_M of the loop algebra (?)_M,a kind of multi-componentintegrable coupling of the above Guo's hierarchy is presented, which has fivepotential functions.