Constructions Of Some Integrable Soliton Hierarchies Based On Lie Algebras

Constructing integrable soliton equations is one of the central problems in soliton theory. In this dissertation, under the guidance of AC= BD model, and with the aid of symbolic computation based on Lie algebras, three topics are studied including discrete integrable soliton hierarchies, nonlinear integrable and bi-integrable couplings of the integrable soliton hierarchy, and super integrable hierarchy and its nonlinear super integrable couplings. There are six chapters in this dissertation:1. In Chapter 1, the development and research achievements of soliton theory, mathematical mechanization, exact solutions of soliton equations, integrable systems, integrable couplings and the main work of this dissertation are introduced.2. In Chapter 2, the basic idea and related conclusions of AC= BD model are introduced, as well as its applications to differential equations and soliton theory.3. In Chapter 3, by designing two new discrete spectral problems, two new discrete integrable systems are obtained. Moreover, we also obtain the Hamiltonian structures of the two lattice hierarchies by means of the discrete trace identity.4. In Chapter 4, by means of two kinds of Lie algebras with the forms of blocks, the nonlinear integrable and bi-integrable couplings of an integrable Hamiltonian system are obtained. Moreover, Hamiltonian structures of nonlinear integrable and bi-integrable couplings of the integrable Hamiltonian system are furnished by applying the variational identity.5. In Chapter 5, based on two Lie super algebras, a super integrable hierarchy and its non-linear super integrable couplings are established, from which some evolution equations are obtained, including the well-known mKdV equation. Meanwhile, their super Hamiltonian structures are given by employing super trace identity.6. In Chapter 6, we summarize the dissertation and give some topics for further research.