Multi-Hamiltonian structure for the finite defocusing Ablowitz-Ladik equation

TL;DR

This paper develops a multi-Hamiltonian framework for the defocusing Ablowitz-Ladik equation by expressing its Poisson structure through the Caratheodory function, revealing connections to orthogonal polynomials and Toda hierarchies.

Contribution

It introduces a family of compatible Poisson brackets forming a multi-Hamiltonian structure and links the Ablowitz-Ladik and Toda hierarchies via Geronimus relations.

Findings

Established a multi-Hamiltonian structure for the Ablowitz-Ladik equation.

Connected orthogonal polynomial relations to integrable hierarchies.

Demonstrated algebraic and symplectic mappings between different integrable systems.

Abstract

We study the Poisson structure associated to the defocusing Ablowitz-Ladik equation from a functional-analytical point of view, by reexpressing the Poisson bracket in terms of the associated Caratheodory function. Using this expression, we are able to introduce a family of compatible Poisson brackets which form a multi-Hamiltonian structure for the Ablowitz-Ladik equation. Furthermore, we show using some of these new Poisson brackets that the Geronimus relations between orthogonal polynomials on the unit circle and those on the interval define an algebraic and symplectic mapping between the Ablowitz-Ladik and Toda hierarchies.