Phase Transition for Discrete Nonlinear Schrödinger Equation in Three and Higher Dimensions

Abstract

We analyze the thermodynamics of the focusing discrete nonlinear Schrödinger equation in dimensions \(d\geqslant 3\) with general power nonlinearity \(p>1\), under a model with two parameters that are inverse temperature and the nonlinearity strength. We prove the existence of the limiting free energy of the associated invariant Gibbs measure and analyze the phase diagram for general d, p. We prove the existence of a continuous phase transition curve that divides the parametric plane into two regions involving the appearance or non-appearance of solitons. Appropriate upper and lower bounds for the curve are constructed that match the result in Chatterjee and Kirkpatrick (Commun Pure Appl Math 65(5):727–757, 2012) a one-sided asymptotic limit. We also look at the typical behavior of a function from the Gibbs measure for parts of the phase diagram.