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Thomas–Fermi Approximation for Coexisting Two Component Bose–Einstein Condensates and Nonexistence of Vortices for Small Rotation

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Abstract

We study minimizers of a Gross–Pitaevskii energy describing a two- component Bose–Einstein condensate confined in a radially symmetric harmonic trap and set into rotation. We consider the case of coexistence of the components in the Thomas–Fermi regime, where a small parameter \({\varepsilon}\) conveys a singular perturbation. The minimizer of the energy without rotation is determined as the positive solution of a system of coupled PDEs, for which we show uniqueness. The limiting problem for \({\varepsilon =0}\) has degenerate and irregular behavior at specific radii, where the gradient blows up. By means of a perturbation argument, we obtain precise estimates for the convergence of the minimizer to this limiting profile, as \({\varepsilon}\) tends to 0. For low rotation, based on these estimates, we can show that the ground states remain real valued and do not have vortices, even in the region of small density.

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Authors and Affiliations

  1. CNRS UMR 8100, Laboratoire de Mathématiques de Versailles, Université de Versailles Saint-Quentin, 45 avenue des Etats-Unis, 78035, Versailles Cédex, France

    Amandine Aftalion & Benedetta Noris

  2. Department of Mathematics and Applied Mathematics, University of Crete, Voutes University Campus, 71003, Heraklion, Crete, Greece

    Christos Sourdis

Authors
  1. Amandine Aftalion
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  2. Benedetta Noris
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  3. Christos Sourdis
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Correspondence to Amandine Aftalion.

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Communicated by L.Caffarelli

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Aftalion, A., Noris, B. & Sourdis, C. Thomas–Fermi Approximation for Coexisting Two Component Bose–Einstein Condensates and Nonexistence of Vortices for Small Rotation. Commun. Math. Phys. 336, 509–579 (2015). https://doi.org/10.1007/s00220-014-2281-9

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  • DOI: https://doi.org/10.1007/s00220-014-2281-9

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