Rotating 2N-vortex solutions to the Gross-Pitaevskii equation on S 2

We establish the existence of rotating solutions to the Gross-Pitaevskii equation \documentclass[12pt]{minimal}\begin{document}$iU_t=\Delta U + \frac{1}{\varepsilon ^2}(1-|U|^2)U$\end{document} i U t = Δ U + 1 ɛ 2 ( 1 − | U | 2 ) U posed on S 2, that is for \documentclass[12pt]{minimal}\begin{document}$U:S^2\times \mathbb {R}\rightarrow \mathbb {C}.$\end{document} U : S 2 × R → C . 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.