TY - JOUR

T1 - Existence, stability, and scattering of bright vortices in the cubic-quintic nonlinear Schrödinger equation

AU - Caplan, R. M.

AU - Carretero-González, R.

AU - Kevrekidis, P. G.

AU - Malomed, B. A.

N1 - Funding Information:
The authors thank Juan Belmonte and Jesús Cuevas for providing useful insights. RCG gratefully acknowledges support from NSF-DMS-0806762 . PGK gratefully acknowledges support from NSF-DMS-0349023 (CAREER) and NSF-DMS-0806762 , as well as from the Alexander von Humboldt Foundation.

PY - 2012/3

Y1 - 2012/3

N2 - 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.

AB - 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.

KW - Modulational instability

KW - Nonlinear Schrödinger equation

KW - Soliton collisions

KW - Variational approximation

KW - Vortices

UR - http://www.scopus.com/inward/record.url?scp=84859428549&partnerID=8YFLogxK

U2 - 10.1016/j.matcom.2010.11.019

DO - 10.1016/j.matcom.2010.11.019

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AN - SCOPUS:84859428549

SN - 0378-4754

VL - 82

SP - 1150

EP - 1171

JO - Mathematics and Computers in Simulation

JF - Mathematics and Computers in Simulation

IS - 7

ER -