Energy estimates and symmetry breaking in attractive Bose–Einstein condensates with ring-shaped potentials

Guo, Yujin; Zeng, Xiaoyu; Zhou, Huan-Song

doi:10.1016/j.anihpc.2015.01.005

Guo, Yujin ; Zeng, Xiaoyu ; Zhou, Huan-Song

Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 3, pp. 809-828 Cet article a éte moissonné depuis la source Numdam

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Résumé

This paper is concerned with the properties of $L^{2}$ -normalized minimizers of the Gross–Pitaevskii (GP) functional for a two-dimensional Bose–Einstein condensate with attractive interaction and ring-shaped potential. By establishing some delicate estimates on the least energy of the GP functional, we prove that symmetry breaking occurs for the minimizers of the GP functional as the interaction strength $a > 0$ approaches a critical value $a^{⁎}$ , each minimizer of the GP functional concentrates to a point on the circular bottom of the potential well and then is non-radially symmetric as $a ↗ a^{⁎}$ . However, when $a > 0$ is suitably small we prove that the minimizers of the GP functional are unique, and this unique minimizer is radially symmetric.

Zbl

DOI : 10.1016/j.anihpc.2015.01.005

Classification : 35J60, 35Q40, 46N50
Keywords: Nonlinear elliptic equation, Constrained minimization, Gross–Pitaevskii functional, Bose–Einstein condensates, Attractive interactions, Ring-shaped potential

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     author = {Guo, Yujin and Zeng, Xiaoyu and Zhou, Huan-Song},
     title = {Energy estimates and symmetry breaking in attractive {Bose{\textendash}Einstein} condensates with ring-shaped potentials},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {809--828},
     year = {2016},
     publisher = {Elsevier},
     volume = {33},
     number = {3},
     doi = {10.1016/j.anihpc.2015.01.005},
     zbl = {1341.35053},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2015.01.005/}
}

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Guo, Yujin; Zeng, Xiaoyu; Zhou, Huan-Song. Energy estimates and symmetry breaking in attractive Bose–Einstein condensates with ring-shaped potentials. Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 3, pp. 809-828. doi: 10.1016/j.anihpc.2015.01.005

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