Fractional discrete vortex solitons

TL;DR

This paper investigates the existence and stability of nonlinear discrete vortex solitons in a square lattice with a fractional Laplacian, revealing that stability domains expand at lower power levels as the fractional exponent decreases.

Contribution

It introduces a fractional discrete Laplacian into the study of vortex solitons, demonstrating extended stability regions and long-range coupling effects.

Findings

Stability domains increase as the fractional exponent decreases.

Long-range coupling decreases faster than exponential with distance.

Stability is independent of topological charge and pattern distribution.

Abstract

We examine the existence and stability of nonlinear discrete vortex solitons in a square lattice when the standard discrete Laplacian is replaced by a fractional version. This creates a new, effective site-energy term, and a coupling among sites, whose range depends on the value of the fractional exponent $\alpha$, becoming effectively long-range at small $\alpha$ values. At long-distance, it can be shown that this coupling decreases faster than exponential: $\sim \exp(- |{\bf n}|)/\sqrt{|\bf{n}|}$. In general, we observe that the stability domain of the discrete vortex solitons is extended to lower power levels, as the $\alpha$ coefficient diminishes, independently of their topological charge and/or pattern distribution.