Symmetries And Lie Algebraic Structures For The Semi-discrete AKNS System

In this dissertation we mainly investigate symmetries and their Lie algebraic struc-tures for some classical integrable semi-discrete systems, which are·four-potential Ablowitz-Ladik hierarchies.·semi-discrete AKNS hierarchies.·integrable semi-discrete nonlinear Schr¨odinger equation.·semi-discrete mKdV equation.We first investigate symmetries of the isospectral and non-isospectral four-potentialAblowitz-Ladik hierarchies. We derive isospectral and non-isospectral equation hierarchiesfrom the four-potential Ablowitz-Ladik spectral problem, and we express them in theform of un,t = LmH(0), where m is an arbitrary integer (instead of a nature number)and L is the recursion operator. Then by means of the zero-curvature representationsof the isospectral and non-isospectral ?ows, we construct two sets of symmetries for theisospectral equation hierarchy as well as non-isospectral equation hierarchy, respectively.The symmetries, respectively, form two centerless Kac-Moody-Virasoro algebras. Therecursion operator L is proved to be hereditary and a strong symmetry for the isospectralequation hierarchy. We also find a strong symmetry operator for each non-isospectral four-potential Ablowitz-Ladik equation. Besides, we make clear for the relation between four-potential and two-potential Ablowitz-Ladik hierarchies together with their symmetriesand algebraic structures. The even order members in the four-potential Ablowitz-Ladikhierarchies together with their symmetries and algebraic structures can be reduced totwo-potential case. The reduction keeps invariant for the algebraic structures and therecursion operator for two potential case becomes L2.Then we consider symmetries for some systems which are related to the Ablowitz-Ladik hierarchy. We derive symmetries for the semi-discrete AKNS hierarchy, integrablesemi-discrete nonlinear Schro¨dinger hierarchy and semi-discrete mKdV hierarchy. Byvirtue of a central-di?erence discretization for the second order derivative in the con-tinuous NLS equation, we present semi-discrete AKNS isospectral ?ows which consist of positive as well as negative order two-potential AL isospectral flows, non-isospectralflows and their recursion operator, respectively. In continuous limit these flows go to thecontinuous AKNS ?ows and the recursion operator goes to the square of the AKNS re-cursion operartor. These semi-discrete AKNS flows form a Lie algebra which plays a keyrole in constructing symmetries and their algebraic structures for both the semi-discreteAKNS hierarchy and its reduction cases. Structures of the obtained algebras are dif-ferent from those in continuous cases which usually are centerless Kac-Moody-Virasorotype. These algebra deformations are explained through continuous limit and"degree"interms of lattice spacing parameter h. As reduced cases, the vector form of the integrablesemi-discrete nonlinear Schro¨dinger hierarchy and semi-discrete mKdV hierarchy can beobtained under the reduction of R_n = -εQ*n and R_n = -εQ_n, respectively. However,for the scalar form of the integrable semi-discrete nonlinear Schro¨dinger hierarchy, theirsymmetries and Lie algebra must guarantee the closeness of algebraic structure. Besides,we construct another semi-discrete AKNS hierarchy and discuss reduction cases, whichalso go to the continuous AKNS system and reduction system under continuous limit.Finally, we discuss the bi-Hamiltonian structures and its continuous limit of the semi-discrete AKNS hierarchy. Using the multi-Hamiltonian structures and implectic operatorθand symplectic operator J of the two-potential Ablowitz-Ladik hierarchy, we presentseveral implectic-symplectic factorization of the recursion operator for the semi-discreteAKNS hierarchy with di?erent linear combinations ofθand J. This enables us to discussbi-Hamiltonian structures of the semi-discrete AKNS hierarchy. In continuous limit thesebi-Hamiltonian structures go to the bi-Hamiltonian structure of the continuous AKNSsystem.