Symbolic Computations Based On The Lie Algebras And Its Application To Integrable Systems

In this paper, we discuss the integrable coupling, self-consistent sources and con-servation laws of the integrable and super integrable systems based on symbolic com-putations on the Lie algebras. The fractional integrable and super integrable systems and algebro-geometric solution of the soliton equations are also discussed. The main contents of this paper are divided into the following four parts:1. We study the integrable couplings of three integrable equation hierarchies by using different methods. Constructing a new spectral problem by extending the loop algebra, the integrable coupling and its Hamiltonian structure of coupled mKdV are given by using Tu format; Starting from the Lie algebra, nonlinear integrable coupling of Guo hierarchy is constructed by expanding the spectrum matrix, using the variational identity to generate its Hamiltonian structure; A new explicit Lie algebra is introduced for which the nonlinear integrable coupling and its Hamiltonian structure of the soliton hierarchy are obtained. Finally, based on an enlarged matrix Lie superalgebra, the nonlinear super integrable coupling and its super Hamiltonian structure are furnished by supertrace identity. As its reduction, we gain the nonlinear integrable coupling of the classical integrable Kaup-Newell hierarchy.2. Nonlinear integrable coupling of the Li hierarchy is given by using the method of constructing integrable coupling, then we consider the properties of nonlinear integrable hierarchy, such as the self-consistent sources and conservation laws using the theory of source. From super Tu integrable hierarchy based on loop Lie superalgebra, the conservation laws of super Tu hierarchy with self-consistent sources is derived using the method of source.3. Based on the theory of fractional derivatives and integrals, the fractional inte-grable coupling and its Hamiltonian structure of Kaup-Newell hierarchy are obtained. By employing the fractional supertrace identity, we derive the fractional super Broer-Kaup-Kupershmidt hierarchy and its fractional super Hamiltonian structure. Then, we present a fractional nonlinear super integrable coupling and its fractional super Hamiltonian structure.4. The algebro-geometric solutions of soliton equations are discussed. Staring from a new spectral problem, a hierarchy of the generalized Kaup-Newell soliton equa-tions and its Hamiltonian structure are derived. The reduced generalized Kaup-Newell soliton equations are decomposed into systems of solvable ordinary differential equa-tions. The appropriate elliptic coordinates and Abel-Jacobi coordinates are introduced to straighten the flows, from which the algebro-geometric solutions of equations are obtained in terms of the Riemann theta functions.