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 Schrödinger equation in one dimension. We compute the generalized free energy of this model and determine the GGE averaged fields and their 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 eigenvalues of COE.

Large Deviations for Ablowitz-Ladik lattice, and the Schur flow

“…Remark 4.2. Our definition of Free Energy is different from the one used in [25,29]. Indeed, in virtue of Varadhan's Lemma [14, Theorem 1.2.1], we have…”

“…From this large deviations principle we are able to deduce that µ N pEq converges almost surely as N goes to infinity. Moreover, following [20,25,29], we are able to characterize the measure ν V β in terms of the equilibrium measure of the Circular beta ensemble at high temperature [21,22]. More precisely, consider the functional µ Þ Ñ f V β pµq given, for any µ P PpTq absolutely continuous with respect to Lebesgue measure and with density dµ dθ , by…”

“…As in [25,29], we introduce the Generalized Gibbs ensemble for the Ablowitz-Ladik lattice, namely the following probability measure on the phase space D N dP V,β AL,N pα 1 , . .…”

“…Thus, we obtain a unique characterization of the measure ν V β . In [25,29], the authors considered the GGE (1) with polynomial potential, and they were able to prove Theorem (1.5) for this particular class of potentials using a transfer operator technique. In this sense, we generalize their result, extending it to the class of continuous potentials.…”

“…To give a wider overview of the relevant literature, we mention that, H. Spohn in [29], applying the theory of generalized hydrodynamics [12], argues that the correlation functions of the Ablowitz-Ladik lattice with respect to the GGE (1) show a ballistic behaviour. As we already mentioned, in [25] the authors rigorously proved Theorem 1.5 for polynomial potential V pzq.…”