# Local minimizers with vortex filaments for a Gross-Pitaevsky functional

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

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

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topJerrard, 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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- A. Montero, P. Sternberg, and W. Ziemer, Local minimizers with vortices to the Ginzburg-Landau system in 3-d. Comm. Pure Appl. Math57 (2004) 99–125.
- 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.
- T. Rivière, Line vortices in the $U\left(1\right)$-Higgs model. Cont. Opt. Calc. Var.1 (1996) 77–167.
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