Dynamics of Ginzburg-Landau and Gross-Pitaevskii vortices on manifolds

Ko-Shin Chen; Peter Sternberg

doi:10.3934/dcds.2014.34.1905

Article Contents

2014, Volume 34, Issue 5: 1905-1931. Doi: 10.3934/dcds.2014.34.1905

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Dynamics of Ginzburg-Landau and Gross-Pitaevskii vortices on manifolds

Ko-Shin Chen^1, and
Peter Sternberg^1,

1.
Department of Mathematics, Indiana University, Bloomington, IN 47405

Received: November 2012

Revised: June 2013

Published: May 2014

Abstract / Introduction Related Papers Cited by

Abstract

We consider the dissipative heat flow and conservative Gross-Pitaevskii dynamics associated with the Ginzburg-Landau energy \begin{equation*} E_\varepsilon(u) = \int_{\mathcal M} \frac{|\nabla_g u|^2}{2} + \frac{(1-|u|^2)^2}{4\varepsilon^2} dv_g \end{equation*} posed on a Riemannian $2$-manifold $\mathcal{M}$ endowed with a metric $g$. In the $ε \to 0$ limit, we show the vortices of the solutions to these two problems evolve according to the gradient flow and Hamiltonian point-vortex flow respectively, associated with the renormalized energy on $\mathcal{M}.$ For the heat flow, we then specialize to the case where $\mathcal{M}=S^2$ and study the limiting system of ODE's and establish an annihilation result. Finally, for the Ginzburg-Landau heat flow on $S^2$, we derive some weighted energy identities.

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Mathematics Subject Classification: Primary: 35R01, 35K15.

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