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ESAIM: Control, Optimisation and Calculus of Variations (1)

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Local minimizers with vortex filamentsfor aGross-Pitaevsky functional
Robert L. Jerrard
Journal:

ESAIM: Control, Optimisation and Calculus of Variations / Volume 13 / Issue 1 / January 2007

Published online by Cambridge University Press:

14 February 2007, pp. 35-71

Print publication:

January 2007
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This paper gives a rigorous derivationof a functional proposed by Aftalion and Rivière [Phys. Rev. A64 (2001) 043611] to characterize the energy of vortex filamentsin a rotationally forced Bose-Einstein condensate. Thisfunctional is derived as a Γ-limitof 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 suchlocal minimizer, the Gross-Pitaevsky functionalhas a local minimizer that is nearby (in a suitable sense) whenever a scalingparameter is sufficiently small.