Existence, Stability, and Dynamics of Bright Vortices in the Cubic-Quintic Nonlinear Schrödinger Equation

Details Existence, Stability, and Dynamics of Bright Vortices in the Cubic-Quintic Nonlinear Schrödinger Equation Existence, Stability, and Dynamics of Bright Vortices in the Cubic-Quintic Nonlinear Schrödinger Equation

2009-10-30

arxiv.org/abs/0910.5758v3

Nonlinear Sciences

Pattern Formation and Solitons

18 pages, 17 figures, accepted to Math. Comm. Sim

Scientific paper

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.

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