Comptes Rendus
Partial Differential Equations
Properties of a single vortex solution in a rotating Bose Einstein condensate
Comptes Rendus. Mathématique, Volume 336 (2003) no. 9, pp. 713-718.

In this Note, we study the properties of the line energy for a vortex γ in a Bose Einstein condensate rotating at velocity Ω. The global minimizer is either the vortex free solution or U vortices which exist only for Ω bigger than a critical value. For all values of Ω, we prove the existence of an S type vortex, which is a critical point of the line energy, observed in the experiments. We also prove uniqueness of the minimizer for almost every Ω and a monotonicity property of the curve γ with respect to Ω. The proofs rely on a related isoperimetric problem.

Dans cette Note, nous étudions les propriétés de l'énergie de ligne pour un vortex γ dans un condensat de Bose Einstein en rotation à la vitesse Ω. Nous prouvons que, pour tout Ω, il existe un vortex de type S, qui est un point critique de l'énergie, mais jamais un minimiseur. Le minimiseur global est soit la solution sans vortex soit un vortex en U, qui n'existe que pour Ω plus grand qu'une valeur critique. Nous prouvons également l'unicité des minimiseurs pour presque tout Ω et une propriété de monotonie des courbes γ par rapport à Ω. Les preuves reposent sur un problème de type isopérimétrique.

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DOI: 10.1016/S1631-073X(03)00166-3
Amandine Aftalion 1; Robert L. Jerrard 2

1 Laboratoire Jacques-Louis Lions, Université Paris 6, 175, rue du Chevaleret, 75013 Paris, France
2 Department of Mathematics, 100 St George St, University of Toronto, Toronto M5S 3G3, Canada
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Amandine Aftalion; Robert L. Jerrard. Properties of a single vortex solution in a rotating Bose Einstein condensate. Comptes Rendus. Mathématique, Volume 336 (2003) no. 9, pp. 713-718. doi : 10.1016/S1631-073X(03)00166-3. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00166-3/

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[2] A. Aftalion; R.L. Jerrard Shape of vortices for a rotating Bose–Einstein condensate, Phys. Rev. A, Volume 66 (2002), p. 023611

[3] A. Aftalion; T. Riviere Vortex energy and vortex bending for a rotating Bose–Einstein condensate, Phys. Rev. A, Volume 64 (2001), p. 043611

[4] E. Giusti Minimal Surfaces and Functions of Bounded Variation, Birkhäuser, Boston, 1984

[5] R.L. Jerrard, in preparation

[6] K. Madison; F. Chevy; W. Wohlleben; J. Dalibard Vortex formation in a stirred Bose–Einstein condensate, Phys. Rev. Lett., Volume 84 (2000), pp. 806-809

[7] K.W. Madison; F. Chevy; W. Wohlleben; J. Dalibard Vortex lattices in a stirred Bose–Einstein condensate, J. Mod. Opt., Volume 47 (2000), p. 2715

[8] C. Raman; J.R. Abo-Shaeer; J.M. Vogels; K. Xu; W. Ketterle Vortex nucleation in a stirred Bose–Einstein condensate, Phys. Rev. Lett., Volume 87 (2001), p. 210402

[9] P. Rosenbuch; V. Bretin; J. Dalibard Dynamics of a single vortex line in a Bose–Einstein condensate, Phys. Rev. Lett., Volume 89 (2002), p. 200403

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