Existence, Stability, and Dynamics of Bright Vortices in the Cubic-Quintic Nonlinear Schr\"odinger Equation

TL;DR

This paper investigates the existence, stability, and interactions of vortex solitons in the 2D cubic-quintic nonlinear Schrödinger equation using semi-analytical and numerical methods, providing new insights into their stability bounds and collision dynamics.

Contribution

It introduces a semi-analytical approach to predict vortex soliton properties and stability, clarifies existence bounds, and explores vortex collision outcomes in the cubic-quintic NLS.

Findings

Predicted vortex stability bounds closely match numerical results.

Identified critical frequencies for vortex instability.

Characterized collision outcomes between vortices with different charges.

Abstract

We revisit the topic of the existence and azimuthal modulational stability of solitary vortices (alias vortex solitons) in the two-dimensional (2D) cubic-quintic nonlinear Schr{\"o}dinger equation. We develop a semi-analytical approach, assuming that the vortex soliton is relatively narrow, and thus splitting the full 2D equation into radial and azimuthal 1D equations. A variational approach is used to predict the radial shape of the vortex soliton, using the radial equation, yielding results very close to those obtained from numerical solutions. Previously known existence bounds for the solitary vortices are recovered by means of this approach. The 1D azimuthal equation of motion is used to analyze the modulational instability of the vortex solitons. The semi-analytical predictions -- in particular, that for the critical intrinsic frequency of the vortex soliton at the instability border -- are compared to systematic 2D simulations. We also compare our findings to those reported in earlier works, which featured some discrepancies. We then perform a detailed computational study of collisions between stable vortices with different topological charges. Borders between elastic and destructive collisions are identified.