Existence, stability, and scattering of bright vortices in the cubic-quintic nonlinear Schrödinger equation

R. M. Caplan, R. Carretero-González*, P. G. Kevrekidis, B. A. Malomed

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

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 Schrödinger equation. We develop a semi-analytical approach, assuming that the vortex soliton is relatively narrow, which allows one to effectively split 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, 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.

Original languageEnglish
Pages (from-to)1150-1171
Number of pages22
JournalMathematics and Computers in Simulation
Volume82
Issue number7
DOIs
StatePublished - Mar 2012

Funding

FundersFunder number
NSF-DMS-0349023NSF-DMS-0806762
National Science Foundation0806762
Alexander von Humboldt-Stiftung

    Keywords

    • Modulational instability
    • Nonlinear Schrödinger equation
    • Soliton collisions
    • Variational approximation
    • Vortices

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