**Discrete Contin. Dyn. Syst. 26:1**, 151-196 (2010)

## The algebro-geometric initial value problem for the Ablowitz-Ladik hierarchy

### F. Gesztesy, H. Holden, J. Michor, and G. Teschl

**Keywords:**
Ablowitz-Ladik hierarchy, complex-valued solutions, initial value
problem.

**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.

**MSC:** Primary 37K10, 37K20, 47B36; Secondary 35Q58, 37K60.

**ESI Preprint** 1929

**arXiv:0706.3370**