Functional representation of the Ablowitz-Ladik hierarchy

The Ablowitz-Ladik hierarchy (ALH) is considered in the framework of the inverse-scattering approach. After establishing the structure of solutions of the auxiliary linear problems, the ALH, which has been originally introduced as an infinite system of differential-difference equations, is presented as a finite system of difference-functional equations. The representation obtained, when rewritten in terms of Hirota's bilinear formalism, is used to demonstrate relations between the ALH and some other integrable systems, the Kadomtsev-Petviashvilli hierarchy in particular.