The periodic defocusing Ablowitz-Ladik equation and the geometry of Floquet CMV matrices

TL;DR

This paper links the periodic defocusing Ablowitz-Ladik equation to Floquet CMV matrices, demonstrating its integrability through a group-theoretic approach and solving related Hamiltonian flows.

Contribution

It introduces a novel geometric framework connecting the Ablowitz-Ladik equation with Floquet CMV matrices using Poisson Lie groups and establishes its Liouville integrability.

Findings

The equation is an isospectral deformation of Floquet CMV matrices.

Floquet CMV matrices form a Coxeter dressing orbit of a loop group.

The integrability is proven via action-angle variables and Riemann-Hilbert problems.

Abstract

In this work, we show that the periodic defocusing Ablowitz-Ladik equation can be expressed as an isospectral deformation of Floquet CMV matrices. We then introduce a Poisson Lie group whose underlying group is a loop group and show that the set of Floquet CMV matrices is a Coxeter dressing orbit of this Poisson Lie group. By using the group-theoretic framework, we establish the Liouville integrability of the equation by constructing action-angle variables, we also solve the Hamiltonian equations generated by the commuting flows via Riemann-Hilbert factorization problems.