Study On Some Topics To Integrable Systems And Super Integrable Systems

In this paper, the integrable coupling of integrable hierarchy, the conservation laws and self-consistent sources of the super integrable systems, fractional super integrable systems and the application of Bell polynomial are mainly discussed.Chapter1is an introduction to review the theoretical background and development of integrable systems, solution of the soliton equation and symbolic computation. The main works of this dissertation are also illustrated.Chapter2is devoted to study two different kinds of integrable couplings of inte-grable hierarchies. From the constructed new enlarged Lie algebra and its corresponding loop algebra, new isospectral problems are discussed, which give rise integrable hierar-chies, and then the integrable couplings of the integrable systems and the corresponding Hamiltonian structures are obtained; From coupled Lie algebras, a coupling integrable couplings of an equation hierarchy is constructed and its corresponding Hamiltonian structure is also obtained by quadratic-form identity.Chapter3concentrates on studying the properties of super integrable hierarchy, such as the self-consistent sources and infinitely conservation laws. From super-KN integrable hierarchies based on loop super Lie algebra, we consider the properties of super-KN integrable hierarchy, such as the self-consistent sources and infinitely con-servation laws using the theory of source. The same method can be used to get the self-consistent sources and the infinitely conservation laws of super-JM integrable hier-archy.Chapter4discusses the generating of the fractional super hierarchy using the modified Riemann-Liouville derivative, based on the theory of fractional derivatives and integrals. Using the theory of the generating of fractional super systems, we generate the fractional super AKNS hierarchy and obtained its fractional super Hamiltonian structure with the help of fractional supertrace identity.Chapter5deals with Bell polynomials and related integrabilities. Via Bell poly-nomial approach and symbolic computation, the extended Korteweg-de Vries equation is transformed into two kinds of bilinear equations by choosing different coefficients. N-soliton solutions, bilinear Backlund transformation, Lax pair. Darboux covariant Lax pair and infinite conservation laws are constructed to reveal the integrability of the equation. At the same time, based on Hirota bilinear method and Riemann theta function, its quasi-periodic wave solution is also presented.