The Integrable Systems And Integrable Coupling Of Solitons Equations

The major contents in this paper include: the formulation of the expandingintegrable model of the (2+1)-dimensional integrable equation hierarchy, theformulation of multi-component integrableeqution hierarchies and the expandingintegrable models, two high-dimensional Lie algebra and their applications. Inthe second chapter, firstly based on a old Lie algebra, a six dimensional liealgebras are constructed through the linear combination, then the correspondingloop algebra is obtained. From the loop algebra a generalized isospectralproblems are designed. Using Tu scheme, an expanding integrable model of(2+1)-dimensional TC hierarchy is worked out. Secondly, on the base of amulti-component loop algebra, an expanding integrable model of multi-component(2+1)-dimensional GJ hierarchy is worked out byuse of (2+1)-dimensional zerocurvature equation and Tu scheme. Thirdly, a multi-component form of equation(8)in reference[40]is worked out, then its Hamiltonian structure is presented byuse of the extended trace identity. Nextly, the expanding integrable model ofmulti-component integrable equation hierarchy is obtianed. The third chaptermain research is two new high-dimensional Lie algebra and its applications.Firstly, we expand a finite dimensional Lie algebra into a higher-dimensionalone. By making use of the later and its corresponding loop algebra, the expandingintegrable model of NLS-MKdV hierarchy is worked out. Secondly, A high-dimensional Lie algebra and its corresponding loop algebra is presented. As itsapplication, an expanding integrable model of AKNS hierarchy is worked out. Byemploying the quadratic-form identity, its Hamiltonian structure is obtained.Furthermore, a high-dimensional Lie algebra is decomposed. From its decom-position a expanding integrable model of the discrete lattice integrable systemin reference[12] is worked out. Then, a simple method is proposed. By which, boththe continuously expanding integrable model and the discrete expanding integrable model are obtained.