Hydrodynamic equations for the Ablowitz-Ladik discretization of the nonlinear Schroedinger equation

TL;DR

This paper derives hydrodynamic equations for the Ablowitz-Ladik discretization of the nonlinear Schrödinger equation, connecting integrable lattice models with random matrix ensembles and extending generalized hydrodynamics to discrete systems.

Contribution

It computes the generalized free energy and derives hydrodynamic equations for the Ablowitz-Ladik model, linking it to the circular unitary ensemble and related integrable systems.

Findings

Hydrodynamic equations match known patterns from integrable many-body systems.

The model's generalized free energy relates to the circular unitary matrix ensemble.

Discretized mKdV equation connects to the beta Jacobi log gas.

Abstract

Ablowitz and Ladik discovered a discretization which preserves the integrability of the nonlinear Schroedinger equation in one dimension. We compute the generalized free energy of this model and determine the GGE averaged fields and currents. They are linked to the mean-field circular unitary matrix ensemble (CUE). The resulting hydrodynamic equations follow the pattern already known from other integrable many-body systems. Studied is also the discretized modified Korteweg-de-Vries equation which turns out to be related to the beta Jacobi log gas.