Hydrodynamic equations for the Ablowitz–Ladik discretization of the nonlinear Schrödinger equation
Herbert Spohn
Abstract
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.
Topics & Concepts
DiscretizationIntegrable systemMathematicsNonlinear systemMathematical physicsKorteweg–de Vries equationMathematical analysisNonlinear Schrödinger equationDimension (graph theory)Lax pairSchrödinger equationPhysicsQuantum mechanicsPure mathematicsNonlinear Waves and SolitonsCold Atom Physics and Bose-Einstein CondensatesNonlinear Photonic Systems