Symmetry, winding number and topological charge of vortex solitons in discrete-symmetry media

TL;DR

This paper analyzes the properties of discrete vortices in media with discrete rotational symmetry, revealing how symmetry constrains vortex charge and the formation of off-axis singularities, supported by numerical examples in photonic crystal fibers.

Contribution

It provides a theoretical framework linking symmetry to vortex charge restrictions and demonstrates this with numerical simulations in photonic crystal fibers.

Findings

High-charged vortices are constrained by symmetry.

Off-axis singularities are necessary for high charges.

Numerical examples confirm theoretical predictions.

Abstract

We determine the functional behavior near the discrete rotational symmetry axis of discrete vortices of the nonlinear Schr\"odinger equation. We show that these solutions present a central phase singularity whose charge is restricted by symmetry arguments. Consequently, we demonstrate that the existence of high-charged discrete vortices is related to the presence of other off-axis phase singularities, whose positions and charges are also restricted by symmetry arguments. To illustrate our theoretical results, we offer two numerical examples of high-charged discrete vortices in photonic crystal fibers showing hexagonal discrete rotational invariance.