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Existence and stability of standing waves for nonlinear Schrödinger equations with a critical rotational speed

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Abstract

We study the existence and stability of standing waves associated with the Cauchy problem for the nonlinear Schrödinger equation (NLS) with a critical rotational speed and an axially symmetric harmonic potential. This equation arises as an effective model describing the attractive Bose–Einstein condensation in a magnetic trap rotating with an angular velocity. By viewing the equation as NLS with a constant magnetic field and with (or without) a partial harmonic confinement, we establish the existence and orbital stability of prescribed mass standing waves for the equation with mass-subcritical, mass-critical, and mass-supercritical nonlinearities. Our result extends a recent work of Bellazzini et al. (Commun Math Phys 353(1):229–251, 2017), where the existence and stability of standing waves for the supercritical NLS with a partial confinement were established.

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Data sharing is not applicable to this article as no new data were created or analyzed in this study.

Notes

  1. Since we consider the rotation on the \((x_1,x_2)\)-plane, the rotation speed should be compared only with the trapping frequencies in the \(x_1\) and \(x_2\) directions.

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Acknowledgements

This work was supported in part by the European Union’s Horizon 2020 Research and Innovation Programme (Grant agreement CORFRONMAT No. 758620, PI: Nicolas Rougerie). The author would like to express his deep gratitude to his wife—Uyen Cong for her encouragement and support. He also would like to thank the reviewers for their helpful comments and suggestions.

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  1. CNRS, UMPA (UMR 5669), Ecole Normale Supérieure de Lyon, Lyon, France

    Van Duong Dinh

  2. Department of Mathematics, Ho Chi Minh City University of Education, 280 An Duong Vuong, Ho Chi Minh City, Vietnam

    Van Duong Dinh

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Dinh, V.D. Existence and stability of standing waves for nonlinear Schrödinger equations with a critical rotational speed. Lett Math Phys 112, 53 (2022). https://doi.org/10.1007/s11005-022-01549-8

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  • DOI: https://doi.org/10.1007/s11005-022-01549-8

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