Vortex solutions of the discrete Gross-Pitaevskii equation

TL;DR

This paper investigates the stability and dynamics of vortex solutions in the two-dimensional discrete nonlinear Schrödinger model, revealing stability windows and the effects of coupling strength and trapping potentials.

Contribution

It provides systematic methods for continuing vortex states from the continuum and anti-continuum limits, and analyzes their stability in discrete and trapped settings.

Findings

Discrete vortices become unstable beyond a critical coupling.

Vortices are restabilized in the continuum limit.

Stability windows depend on lattice parameters and trapping potentials.

Abstract

In this paper, we consider the dynamical evolution of dark vortex states in the two-dimensional defocusing discrete nonlinear Schroedinger model, a model of interest both to atomic physics and to nonlinear optics. We find that in a way reminiscent of their 1d analogs, i.e., of discrete dark solitons, the discrete defocusing vortices become unstable past a critical coupling strength and, in the infinite lattice, they apparently remain unstable up to the continuum limit where they are restabilized. In any infinite lattice, stabilization windows of the structures may be observed. Systematic tools are offered for the continuation of the states both from the continuum and, especially, from the anti-continuum limit. Although the results are mainly geared towards the uniform case, we also consider the effect of harmonic trapping potentials.