Contemporary Mathematics Volume 446, 2007 Vortex patterns in Bose Einstein condensates Amandine Aftalion This paper is dedicated to Haim Brezis for his 60th birthday ABSTRACT. This paper deals with the mathematical analysis of some features emerging in recent experiments on Bose Einstein condensates which display vortices. We study in particular the shape and location of the vortex lines, in the framework of the Gross Pitaevskii equation. Some of the tools are inspired by the famous book of Bethuel-Brezis-Helein [BBH] on Ginzburg- Landau vortices. The results presented here provide an illustration of how the theory and tools introduced by Bethuel-Brezis-Helein [BBH] for the study of Ginzburg-Landau vor- tices, developped and extended by many authors, can be useful for the analysis of problems arising in today's experimental condensed matter physics. Indeed, our work has been motivated by recent experiments in the Kastler Brossellaboratory of Ecole normale superieure, in the group of Jean Dalibard [BSSD, MCBD, RBD, SBCD]. The results described here have been obtained with various collabora- tors: Tristan Riviere [AR], Bob Jerrard [AJl, AJ2], Ionut Danaila [ADl, AD2], Stan Alama et Lia Bronsard [AAB], Xavier Blanc [AB2, AB3], Xavier Blanc and Jean Dalibard [ABD], Xavier Blanc and Francis Nier [ABN]. All this and more is explained in details in a book which has just appeared [Aft]. 1. Introduction Since the first experimental achievement of Bose Einstein condensation in con- fined alkali gases in 1995 awarded with the Nobel Prize for Physics in 2001 [CW, K], many properties of these systems have been studied experimentally and theoreti- cally, in particular in the Kastler Brossellaboratory. The mathematical description is only starting [Aft]. 1.1. Physical context. The phenomenon of condensation was predicted in 1925 by Einstein, on the basis of a paper by Bose: for a gas of non interacting 1991 Mathematics Subject Classification. 35Jxx, 35Q40, 81V45. Key words and phrases. Bose Einstein condensates, vortices, lattice, Gross Pitaevskii equations. @2007 American Mathematical Society
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