Generalized Gibbs Ensemble of the Ablowitz–Ladik Lattice, Circular $$\beta $$-Ensemble and Double Confluent Heun Equation

Abstract

AbstractWe consider the discrete defocusing nonlinear Schrödinger equation in its integrable version, which is called defocusing Ablowitz–Ladik lattice. We consider periodic boundary conditions with periodNand initial data sampled according to the Generalized Gibbs ensemble. In this setting, the Lax matrix of the Ablowitz–Ladik lattice is a random CMV-periodic matrix and it is related to the Killip-Nenciu Circular$$\beta $$β-ensemble at high-temperature. We obtain the generalized free energy of the Ablowitz–Ladik lattice and the density of states of the random Lax matrix by establishing a mapping to the one-dimensional log-gas. For the Gibbs measure related to the Hamiltonian of the Ablowitz–Ladik flow, we obtain the density of states via a particular solution of the double-confluent Heun equation.