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Finite genus solutions for the Ablowitz-Ladik hierarchy

The question of constructing the finite genus quasiperiodic solutions for the Ablowitz-Ladik hierarchy (ALH) is considered by establishing relations between the ALH and the Fay's identity for the theta-functions. It is shown that using a limiting procedure one can derive from the latter an infinite number of differential identities which can be arranged as an infinite set of differential-difference equations coinciding with the equations of the ALH, and that the original Fay's identity can be rewritten in a form similar to the functional equations representing the ALH which have been derived in the previous works of the author. This provides an algorithm for obtaining some class of quasiperiodic solutions for the ALH, which can be viewed as an alternative to the inverse scattering transform or the algebro-geometrical approach.