The Study About Some Topics Of Integtable System And Its Hamilton Structure

Looking for the integrable coupling and its Hamilton structure are very important topics in the study of integrable system. Around the two themes in this paper, we research the integrable coupling of integrable system and fractional integrable system, as well as the application of the quadratic-form identity, the fractional quadratic-form identity and fractional supertrace identity.The chapter 1 is introduction that review the definition of two kinds of integrability, the development of integrable system and my main work in this paper.In chapter 2, we get the new integrable coupling of C-Kd V hierarchy through combining the method expansion the spectral problem, construct new loop algebras and the method semi-directe sums Lie algebra. Then a series of new equations are generated. Under the new Loop Lie algebras and the quadratic-form identity, we construct bi-Hamilton structures of integrable coupling of C-Kd V hierarchy.The chapter 3 discusses the generating of the fractional integrable coupling of fractional coupled Burgers, based on the theory of fractional calculus and using the method which expand the spectral problem and new loop algebras. Under the fractional quadratic-form identity, the bi-Hamilton structures of the integrable coupling of coupled Burgers hierarchy are obtained.The chapter 4 studies the generating of the fractional super integrable hierarchy using the modified Riemann-Liouville derivative, under the background of the theory of fractional derivatives and integrals. Using the theory of the generating of fractional super systems, we get the fractional super Yang hierarchy and generate its fractional super Hamilton structure with the help of fractional supertrace identity.