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

TL;DR

This paper develops a new algorithm to solve the algebro-geometric initial value problem for the Ablowitz-Ladik hierarchy with complex data, proving global unique solvability and addressing challenges for non-unitary Lax operators.

Contribution

It introduces a novel method for constructing algebro-geometric solutions and solves the inverse spectral problem for general Ablowitz-Ladik Lax operators.

Findings

Proves global in-time unique solvability for a full measure set of initial data.

Develops a new algorithm for stationary algebro-geometric solutions.

Addresses difficulties with non-unitary Lax operators in integrable systems.

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.