Phase Transition for Discrete Non Linear Schr\"odinger Equation in Three and Higher Dimensions

TL;DR

This paper investigates the thermodynamics and phase transitions of the focusing discrete nonlinear Schrödinger equation in three or more dimensions, revealing conditions for soliton formation and analyzing the phase diagram.

Contribution

It establishes the existence of limiting free energy and characterizes the phase transition curve for the model in higher dimensions with general nonlinearity.

Findings

Existence of limiting free energy in the model.

Identification of a continuous phase transition curve.

Bounds for the phase transition curve.

Abstract

We analyze the thermodynamics of the focusing discrete nonlinear Schr\"odinger equation in dimensions $d\ge 3$ with general nonlinearity $p>1$ and under a model with two parameters, representing inverse temperature and strength of the nonlinearity, respectively. We prove the existence of limiting free energy and analyze the phase diagram for general $d,p$. We also 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. We also look at the typical behavior of a function chosen from the Gibbs measure for certain parts of the phase diagram.