F. Gesztesy, H. Holden, J. Michor, and G. Teschl
The algebro-geometric initial value problem for the Ablowitz-Ladik hierarchy
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ABSTRACT.  We discuss the algebro-geometric initial value problem for the Ablowitz-Ladik hierarchy 
with complex-valued initial data and prove unique solvability globally in 
time for a set of initial (Dirichlet divisor) data of full measure. To this effect we develop a new algorithm for constructing stationary complex-valued algebro-geometric solutions of the Ablowitz-Ladik hierarchy, which is of independent interest as it solves the inverse 
algebro-geometric spectral problem for generally non-unitary Ablowitz-Ladik Lax operators, starting from a suitably chosen set of initial divisors of full measure. Combined with an appropriate first-order system of differential equations with respect to time (a substitute for the well-known Dubrovin-type equations), this yields the construction of global algebro-geometric solutions of the time-dependent Ablowitz-Ladik hierarchy. 
The generally non-unitary behavior of the underlying Lax operator associated with general coefficients for the Ablowitz-Ladik hierarchy poses a variety of difficulties that, to the best of our knowledge, are successfully overcome here for the first time. Our approach is not confined to the Ablowitz-Ladik hierarchy but applies generally to 1+1-dimensional completely integrable (discrete and continuous) soliton equations.