Symmetries for the Ablowitz-Ladik Hierarchy: Part II. Integrable Discrete Nonlinear Schrodinger Equations and Discrete AKNS Hierarchy

In the paper, we continue to consider symmetries related to the Ablowitz-Ladik hierarchy. We derive symmetries for the integrable discrete nonlinear Schrodinger hierarchy and discrete AKNS hierarchy. The integrable discrete nonlinear Schrodinger hierarchy is in scalar form and its two sets of symmetries are shown to form a Lie algebra. We also present discrete AKNS isospectral flows, non-isospectral flows and their recursion operator. In continuous limit these flows go to the continuous AKNS flows and the recursion operator goes to the square of the AKNS recursion operator. These discrete AKNS flows form a Lie algebra that plays a key role in constructing symmetries and their algebraic structures for both the integrable discrete nonlinear Schrodinger hierarchy and discrete AKNS hierarchy. Structures of the obtained algebras are different structures from those in continuous cases which usually are centerless Kac-Moody-Virasoro type. These algebra deformations are explained through continuous limit and degree in terms of lattice spacing parameter h.