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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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