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Vortex filament dynamics for Gross-Pitaevsky type equations. (English) Zbl 1170.35318
Summary: We study solutions of the Gross-Pitaevsky equation and similar equations in \(m\geq 3\) space dimensions in a certain scaling limit, with initial data \(u^\varepsilon_0\) for which the Jacobian \(Ju^\varepsilon_0\) concentrates around an (oriented) rectifiable \(m-2\) dimensional set, say \(\Gamma_0\), of finite measure. It is widely conjectured that under these conditions, the Jacobian at later times \(t>0\) continues to concentrate around some codimension 2 submanifold, say \(\Gamma_t\), and that the family \(\{\Gamma_t\}\) of submanifolds evolves by binomial mean curvature flow. We prove this conjecture when \(\Gamma_0\) is a round \(m-2\)-dimensional sphere with multiplicity 1. We also prove a number of partial results for more general inital data.

MSC:
35B25 Singular perturbations in context of PDEs
35Q55 NLS equations (nonlinear Schrödinger equations)
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