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Rotating 2*N*-vortex solutions to the Gross-Pitaevskii equation on *S* ^{2}

### Abstract

We establish the existence of rotating solutions to the Gross-Pitaevskii equationposed on *S* ^{2}, that is for These solutions possess vortices that for all time follow the vortex paths of known “relative equilibria” to the point-vortex problem on the two-sphere in the asymptotic regime ɛ ≪ 1. The approach is variational, based on minimization of the Ginzburg-Landau energy subject to a momentum constraint. We also establish orbital stability within a class of symmetric initial data.

© 2012 American Institute of Physics

Received 24 May 2012
Accepted 11 July 2012

Acknowledgments:
The authors were supported in this research by the National Science Foundation (NSF) through Grant Nos. DMS-0654122 and DMS-1101290. P.S. would also like to thank Paul Newton for generously providing many helpful references and Robert Jerrard for helpful conversations.

Article outline:

I. INTRODUCTION
II. ROTATING SOLUTIONS FOR THE 2*N*-VORTEX PROBLEM
III. NOTATION AND PRELIMINARIES
IV. CONSTRUCTION OF A ROTATING 2-VORTEX SOLUTION
V. 2*N*-VORTEX SOLUTIONS FOR *N* > 1
VI. ORBITAL STABILITY

/content/aip/journal/jmp/53/8/10.1063/1.4739748

http://aip.metastore.ingenta.com/content/aip/journal/jmp/53/8/10.1063/1.4739748

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