We consider a two-dimensional nonlinear Schrödinger equation proposed in Physics to model rotational binary Bose–Einstein condensates. The nonlinearity is a logarithmic modification of the usual cubic nonlinearity. The presence of both the external confining potential and rotating frame makes it difficult to apply standard techniques to directly construct ground states, as we explain in an appendix. The goal of the present paper is to analyze the orbital stability of the set of energy minimizers under mass constraint, according to the relative strength of the confining potential compared to the angular frequency. The main novelty concerns the critical case where these two effects compensate exactly (lowest Landau Level): orbital stability is established by using techniques related to magnetic Schrödinger operators.
Nous considérons une équation de Schrödinger non linéaire en deux dimensions d’espace, introduite en physique pour modéliser les condensats de Bose–Einstein en rotation. La non-linéarité est une modification logarithmique du terme cubique habituel. Les présences conjuguées d’un potentiel confinant et d’un repère tournant font qu’il est difficile d’appliquer les techniques standard dans la construction d’états fondamentaux, comme expliqué en appendice. Le but de ce papier est d’analyser la stabilité orbitale de l’ensemble des minimiseurs d’énergie à masse fixée, selon la valeur relative de la force du potentiel confinant par rapport à la vitesse de rotation. La nouveauté principale concerne le cas critique où les deux effets se compensent exactement (niveau fondamental de Landau) : la stabilité orbitale est démontrée en utilisant des techniques en lien avec les opérateurs de Schrödinger magnétiques.
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Keywords: Nonlinear Schrödinger equation, Bose–Einstein condensate, Harmonic potential, Rotation, Standing waves, Stability, Magnetic Schrödinger operators
Mot clés : Équations de Schrödinger non linéaires, condensation de Bose–Einstein, potentiel harmonique, rotation, états stationnaires, stabilité, opérateur de Schrödinger magnétique
Rémi Carles 1; Van Duong Dinh 2; Hichem Hajaiej 3
CC-BY 4.0
@article{AFST_2023_6_32_1_81_0,
author = {R\'emi Carles and Van Duong Dinh and Hichem Hajaiej},
title = {On stability of rotational {2D} binary {Bose{\textendash}Einstein} condensates},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {81--124},
publisher = {Universit\'e Paul Sabatier, Toulouse},
volume = {Ser. 6, 32},
number = {1},
year = {2023},
doi = {10.5802/afst.1730},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1730/}
}
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Rémi Carles; Van Duong Dinh; Hichem Hajaiej. On stability of rotational 2D binary Bose–Einstein condensates. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 32 (2023) no. 1, pp. 81-124. doi : 10.5802/afst.1730. https://afst.centre-mersenne.org/articles/10.5802/afst.1730/
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