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Comptes Rendus Mathématique
Volume 336, n° 9
pages 713-718 (mai 2003)
Doi : 10.1016/S1631-073X(03)00166-3
Received : 17 Mars 2003 ;  accepted : 17 Mars 2003
Properties of a single vortex solution in a rotating Bose Einstein condensate

Amandine  Aftalion a ,  Robert L.  Jerrard b
aLaboratoire Jacques-Louis Lions, Université Paris 6, 175, rue du Chevaleret, 75013 Paris, France 
bDepartment of Mathematics, 100 St George St, University of Toronto, Toronto M5S 3G3, Canada 

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In this Note, we study the properties of the line energy for a vortex   in a Bose Einstein condensate rotating at velocity  . The global minimizer is either the vortex free solution or   vortices which exist only for   bigger than a critical value. For all values of  , we prove the existence of an   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   and a monotonicity property of the curve   with respect to  . The proofs rely on a related isoperimetric problem. To cite this article: A. Aftalion, R.L. Jerrard, C. R. Acad. Sci. Paris, Ser. I 336 (2003).

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Dans cette Note, nous étudions les propriétés de l'énergie de ligne pour un vortex   dans un condensat de Bose Einstein en rotation à la vitesse  . Nous prouvons que, pour tout  , il existe un vortex de type  , 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  , qui n'existe que pour   plus grand qu'une valeur critique. Nous prouvons également l'unicité des minimiseurs pour presque tout   et une propriété de monotonie des courbes   par rapport à  . Les preuves reposent sur un problème de type isopérimétrique. Pour citer cet article : A. Aftalion, R.L. Jerrard, C. R. Acad. Sci. Paris, Ser. I 336 (2003).




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