Lax pairs for the Ablowitz-Ladik system via orthogonal polynomials on the unit circle

TL;DR

This paper constructs Lax pair representations for the defocusing Ablowitz-Ladik system using CMV matrices, extending the understanding of integrable systems related to orthogonal polynomials on the unit circle.

Contribution

It introduces a novel approach to derive Lax pairs for the Ablowitz-Ladik system via CMV matrices in various settings.

Findings

Lax pairs are explicitly constructed for the periodic case.

Lax pairs are explicitly constructed for the finite case.

Lax pairs are explicitly constructed for the infinite case.

Abstract

Nenciu and Simon found that the analogue of the Toda system in the context of orthogonal polynomials on the unit circle is the defocusing Ablowitz-Ladik system. In this paper we use the CMV and extended CMV matrices, respectively, to construct Lax pair representations for this system in the periodic, finite, and infinite cases.