Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-28T14:15:56.210Z Has data issue: false hasContentIssue false
  • English
  • Français

Local minimizers with vortex filamentsfor aGross-Pitaevsky functional

Published online by Cambridge University Press:  14 February 2007

Robert L. Jerrard
Robert L. Jerrard*
Affiliation:
Math Department, University of Toronto, Toronto, ON M5S 3G3, Canada; rjerrard@math.toronto.edu
Get access

Abstract

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.

Keywords

Gross-PitaevskyvorticesGamma-convergenceThomas-Fermi limitrectifiable currents.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 13 , Issue 1 , January 2007 , pp. 35 - 71
Copyright
© EDP Sciences, SMAI, 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aftalion, A. and Jerrard, R.L., On the shape of vortices for a rotating Bose-Einstein condensate. Phys. Rev. A 66 (2002) 023611. CrossRef
Aftalion, A. and Jerrard, R. L., Properties of a single vortex solution in a rotating Bose-Einstein condensate. C. R. Acad. Sci. Paris Ser. I 336 (2003) 713–718. CrossRef
Aftalion, A. and Rivière, T., Vortex energy and vortex bending for a rotating Bose-Einstein condensate. Phys. Rev. A 64 (2001) 043611. CrossRef
Alberti, G., Baldo, S. and Orlandi, G., Functions with prescribed singularities. J. Eur. Math. Soc. 5 (2003) 275–311. CrossRef
Alberti, G., Baldo, S. and Orlandi, G., Variational convergence for functionals of Ginzburg-Landau type. Indiana Univ. Math J. 54 (2005) 1411–1472. CrossRef
Andre, N. and Shafrir, I., Asymptotic behavior of minimizers for the Ginzburg-Landau functional with weight. I, II. Arch. Rational Mech. Anal. 142 (1998) 45–73, 75–98.
F. Bethuel, H. Brezis and F. Hélein, Ginzburg-Landau Vortices. Birkhauser, New-York (1994).
Brezis, H., Coron, J.M., and Lieb, E.H., Harmonic maps with defects. Comm. Math. Phys. 107 (1986) 649–705. CrossRef
L.C. Evans and R.F. Gariepy, Measure Theory and Fine Properties of Functions. CRC Press, London (1992).
H. Federer, Geometric Measure Theory. Springer-Verlag, Berlin (1969).
M. Giaquinta, G. Modica and J. Soucek, Cartesian Currents in the Calculus of Variations. I, II. Springer-Verlag, New York (1998).
Jerrard, R.L. and Soner, H.M., The Jacobian and the Ginzburg-Landau functional. Cal. Var. 14 (2002) 151–191. CrossRef
Jerrard, R.L., Montero, A., and Sternberg, P., Local minimizers of the Ginzburg-Landau energy with magnetic field in three dimensions. Comm. Math. Phys. 249 (2004) 549–577. CrossRef
Kohn, R.V. and Sternberg, P., Local minimizers and singular perturbations. Proc. Royal Soc. Edin. 111A (1989) 69–84. CrossRef
Lassoued, L. and Mironescu, P., Ginzburg-Landau type energy with discontinuous constraint. J. Anal. Math. 77 (1999) 1–26. CrossRef
Montero, A., Sternberg, P., and Ziemer, W., Local minimizers with vortices to the Ginzburg-Landau system in 3-d. Comm. Pure Appl. Math 57 (2004) 99–125. CrossRef
Raman, C., Abo-Shaeer, J. R., Vogels, J. M., Xu, K. and Ketterle, W., Vortex nucleation in a stirred Bose-Einstein condensate. Phys. Rev. Lett. 87 (2001) 210402. CrossRef
Rivière, T., Line vortices in the $U(1)$ -Higgs model. Cont. Opt. Calc. Var. 1 (1996) 77–167.
Rosenbuch, P., Bretin, V., and Dalibard, J., Dynamics of a single vortex line in a Bose-Einstein condensate. Phys. Rev. Lett. 89 (2002) 200403. CrossRef
E. Sandier and S. Serfaty. A product estimate for Ginzburg-Landau and corollaries. J. Funct. Anal. 211 (2004) 219–244.