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Rotating 2N-vortex solutions to the Gross-Pitaevskii equation on \(S^2\). (English) Zbl 1278.35228

Summary: We establish the existence of rotating solutions to the Gross-Pitaevskii equation \(iU_t=\Delta U + \frac{1}{\varepsilon ^2}(1-|U|^2)U\) posed on \(S^2\), that is for \(U:S^2\times \mathbb{R}\rightarrow \mathbb{C}\). 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 \(\varepsilon \ll 1\). 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}

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
82D50 Statistical mechanics of superfluids
82B26 Phase transitions (general) in equilibrium statistical mechanics
78A60 Lasers, masers, optical bistability, nonlinear optics
49S05 Variational principles of physics
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