Fritz Gesztesy, Helge Holden, Johanna Michor, Gerald Teschl
The Algebro-Geometric Initial Value Problem for the Ablowitz-Ladik Hierarchy
The paper is published: Discrete Contin. Dyn. Syst. 26:1 (2010) 151-196

MSC:

47B37 Operators on special spaces (weighted shifts, operators on sequence spaces, etc.)

47B39 Difference operators, See also {39A70}

14H99 None of the above but in this section

58F07 Completely integrable systems (including systems with an infinite number of degrees of freedom)

58D05 Groups of diffeomorphisms and homeomorphisms as manifolds, See also {22E65, 57S05}

58D25 Equations in function spaces; evolution equations, See also {34Gxx, 35K22, 35R15, 47H15}

35Q58 Other completely integrable equations, See also {58F07}

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 general (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 treatment of general (non-unitary) Lax operators 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 soliton
equations of differential-difference type.
Keywords: Ablowitz--Ladik hierarchy, complex-valued solutions, initial value problem
Notes: second and final version