The generalized point-vortex problem and rotating solutions to the Gross–Pitaevskii equation on surfaces of revolution

Abstract

We study the generalized point-vortex problem and the Gross-Pitaevskii equation on a surface of revolution. We find rotating periodic solutions to the generalized point-vortex problem, which have two two rings of n equally spaced vortices with degrees ±1. In particular we prove the existence of such solutions when the surface is longitudinally symmetric. Then we seek a rotating solution to the Gross-Pitaevskii equation having vortices that follow those of the point-vortex flow for ε sufficiently small.