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Existence, stability, and scattering of bright vortices in the cubic-quintic nonlinear Schrödinger equation. (English) Zbl 1302.78022

Summary: 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.

MSC:

78A60 Lasers, masers, optical bistability, nonlinear optics
35Q55 NLS equations (nonlinear Schrödinger equations)
35B35 Stability in context of PDEs
35Q51 Soliton equations
78A45 Diffraction, scattering
78M20 Finite difference methods applied to problems in optics and electromagnetic theory
78M30 Variational methods applied to problems in optics and electromagnetic theory
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