Functional Representation of the Ablowitz–Ladik Hierarchy. II

In this paper we continue studies of the functional representation of the Ablowitz– Ladik hierarchy (ALH). Using formal series solutions of the zero-curvature condition we rederive the functional equations for the tau-functions of the ALH and obtain some new equations which provide more straightforward description of the ALH and which were absent in our previous paper. These results are used to establish relations between the ALH and the discrete-time nonlinear Schrödinger equations, to deduce the superposition formulae (Fay’s identities) for the tau-functions of the hierarchy and to obtain some new results related to the Lax representation of the ALH and its conservation laws. Using the previously found connections between the ALH and other integrable systems we derive functional equations which are equivalent to the AKNS, derivative nonlinear Schrödinger and Davey–Stewartson hierarchies.