Lie Algebras And Researches Of Integrable System

The major content in this paper include:the formulation of the integrable couplings of the discrete integrable equation hierarchy and the formulation of the continuous integrable equation hierarchy as wall as their expanding integrable system.In general,the two major steps is taken as follows:the first step is to expand the lower dimensional Lie algebras which is constructed or exist into higher dimensional Lie algebras according to the semi-direct thought and structure the corresponding Loop algebras;the second step is to devise the isospectral problem resorting to them and deduce the integrable hierarchy of soliton equations and their expanding integrable system.In the first chapter,historical origins and research summarization of soliton theory together with its research meaning is presented.In the second chapter,a few expanding semi-direct sums based on the fundamental special Lie algebra are presented.By use of the above Lie algebras,three discrete integrable couplings of the new Bargmann type lattice equations hierarchy are obtained respectively.In the third chapter,a high-dimensional Lie algebra G is constructed and decomposed,which produce a series of Lie subalgebras.Making use of one of the subalgebras,a new multi-component Lie algebra F_M and its corresponding Loop algebra F_M are constructed,which is devoted to establishing an isospectral problem.By making use of the zero-curvature equation and the quadratic-form identity,the generalized multi-component nonlinear Schrodinger equation hierarchy and its binary Hamiltonian structure is obtained respectively.Again via expanding the Lie algebra F_M,another two higher-dimensional Lie algebra F_M1 and F_M2 are constructed.As their applications,the binary integrable couplings of the generalized multi-component nonlinear Schrodinger equation hierarchy are worked out.And a bi-Hamiltonian Structure of one of the integrable couplings is worked out resorting to the quadratic-form identity.