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Ablowitz and Ladik discovered a discretization which preserves theintegrability of the nonlinear Schroedinger equation in one dimension. Wecompute the generalized free energy of this model and determine the GGEaveraged fields and currents. They are linked to the mean-field circularunitary matrix ensemble (CUE). The resulting hydrodynamic equations follow thepattern already known from other integrable many-body systems. Studied is alsothe discretized modified Korteweg-de-Vries equation which turns out to berelated to the beta Jacobi log gas.

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