Statistical Mechanics of a Discrete Schr\"odinger Equation with Saturable Nonlinearity

TL;DR

This paper investigates the statistical mechanics of a one-dimensional discrete nonlinear Schrödinger equation with saturable nonlinearity, revealing the persistence of localized excitation phenomena despite the nonlinear character diminishing at higher amplitudes.

Contribution

It extends previous work on cubic DNLS to the saturable case, showing that localized excitations and phase transitions persist despite the nonlinearity becoming more linear at high amplitudes.

Findings

Localized excitations are spontaneously created.

Discontinuity in the partition function indicates phase transition.

Phenomenon persists despite increasing linearity at high amplitudes.

Abstract

We study the statistical mechanics of the one-dimensional discrete nonlinear Schr\"odinger (DNLS) equation with saturable nonlinearity. Our study represents an extension of earlier work [Phys. Rev. Lett. {\bf 84}, 3740 (2000)] regarding the statistical mechanics of the one-dimensional DNLS equation with a cubic nonlinearity. As in this earlier study we identify the spontaneous creation of localized excitations with a discontinuity in the partition function. The fact that this phenomenon is retained in the saturable DNLS is non-trivial, since in contrast to the cubic DNLS whose nonlinear character is enhanced as the excitation amplitude increases, the saturable DNLS in fact becomes increasingly linear as the excitation amplitude increases. We explore the nonlinear dynamics of this phenomenon by direct numerical simulations.