ESAIM: Control, Optimisation and Calculus of Variations

Research Article

Local minimizers with vortex filaments for a Gross-Pitaevsky functional

Jerrard, Robert L.

Math Department, University of Toronto, Toronto, ON M5S 3G3, Canada; rjerrard@math.toronto.edu

Abstract

This paper gives a rigorous derivation of a functional proposed by Aftalion and Rivière [Phys. Rev. A 64 (2001) 043611] to characterize the energy of vortex filaments in a rotationally forced Bose-Einstein condensate. This functional is derived as a Γ-limit of scaled versions of the Gross-Pitaevsky functional for the wave function of such a condensate. In most situations, the vortex filament energy functional is either unbounded below or has only trivial minimizers, but we establish the existence of large numbers of nontrivial local minimizers and we prove that, given any such local minimizer, the Gross-Pitaevsky functional has a local minimizer that is nearby (in a suitable sense) whenever a scaling parameter is sufficiently small.

(Received December 13 2004)

(Revised August 22 2005)

(Online publication February 14 2007)

Key Words:

  • Gross-Pitaevsky;
  • vortices;
  • Gamma-convergence;
  • Thomas-Fermi limit;
  • rectifiable currents.

Mathematics Subject Classification:

  • 35Q40;
  • 35B25;
  • 49Q20
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