Local minimizers with vortex filaments for a Gross-Pitaevsky functional

Robert L. Jerrard

ESAIM: Control, Optimisation and Calculus of Variations (2007)

  • Volume: 13, Issue: 1, page 35-71
  • ISSN: 1292-8119

Abstract

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This paper gives a rigorous derivation of a functional proposed by Aftalion and Rivière [Phys. Rev. A64 (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.

How to cite

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Jerrard, Robert L.. "Local minimizers with vortex filaments for a Gross-Pitaevsky functional." ESAIM: Control, Optimisation and Calculus of Variations 13.1 (2007): 35-71. <http://eudml.org/doc/250012>.

@article{Jerrard2007,
abstract = { This paper gives a rigorous derivation of a functional proposed by Aftalion and Rivière [Phys. Rev. A64 (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. },
author = {Jerrard, Robert L.},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Gross-Pitaevsky; vortices; Gamma-convergence; Thomas-Fermi limit; rectifiable currents.; rectifiable currents; wave function; critical point; energy of vortex filaments in a rotationally forced Bose-Einstein condensate; -limit; Gross-Pitaevsky functional},
language = {eng},
month = {2},
number = {1},
pages = {35-71},
publisher = {EDP Sciences},
title = {Local minimizers with vortex filaments for a Gross-Pitaevsky functional},
url = {http://eudml.org/doc/250012},
volume = {13},
year = {2007},
}

TY - JOUR
AU - Jerrard, Robert L.
TI - Local minimizers with vortex filaments for a Gross-Pitaevsky functional
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2007/2//
PB - EDP Sciences
VL - 13
IS - 1
SP - 35
EP - 71
AB - This paper gives a rigorous derivation of a functional proposed by Aftalion and Rivière [Phys. Rev. A64 (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.
LA - eng
KW - Gross-Pitaevsky; vortices; Gamma-convergence; Thomas-Fermi limit; rectifiable currents.; rectifiable currents; wave function; critical point; energy of vortex filaments in a rotationally forced Bose-Einstein condensate; -limit; Gross-Pitaevsky functional
UR - http://eudml.org/doc/250012
ER -

References

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  3. A. Aftalion and T. Rivière, Vortex energy and vortex bending for a rotating Bose-Einstein condensate. Phys. Rev. A64 (2001) 043611.  
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  17. C. Raman, J. R. Abo-Shaeer, J. M. Vogels, K. Xu and W. Ketterle, Vortex nucleation in a stirred Bose-Einstein condensate. Phys. Rev. Lett.87 (2001) 210402.  
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