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

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.

27 pages, improved and expanded version

References in corpus (5)

• Domain wall dynamics in the Landau--Lifshitz magnet and the classical-quantum correspondence for spin transport
• The collision rate ansatz for the classical Toda lattice
• The classical $\beta$-ensembles with $\beta$ proportional to $1/N$: from loop equations to Dyson's disordered chain
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Cited by in corpus (12)

• Inverse scattering solution of the weak noise theory of the Kardar-Parisi-Zhang equation with flat and Brownian initial conditions
• Anomalous relaxation and hyperuniform fluctuations in center-of-mass conserving systems with broken time-reversal symmetry
• New classical integrable systems from generalized $T\bar{T}$-deformations
• Generalized Gibbs ensemble of the Ablowitz-Ladik lattice, Circular $\beta$-ensemble and double confluent Heun equation
• Large Deviations for Ablowitz-Ladik lattice, and the Schur flow
• Discrete integrable systems and random Lax matrices
• Landau-Lifschitz magnets: exact thermodynamics and transport
• Hyperuniformity in mass transport processes with center-of-mass conservation: Some exact results
• CLT for real beta-ensembles at high temperature
• Particle Scattering and Fusion for the Ablowitz-Ladik Chain
• Hydrodynamics of spin transport in the Haldane-Shastry chain
• Lax random matrices from Calogero systems