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

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