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Strong Instability of Standing Waves for the Nonlinear Schrödinger Equation in Trapped Dipolar Quantum Gases

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Abstract

In this paper, we study the strong instability of standing waves for the nonlinear Schrödinger equation arising in trapped dipolar quantum gases. Two cases are considered: the first when the system is free, the second when a partial/complete harmonic potential is added. In the free case, we present a new argument to prove that the ground state standing waves are strongly unstable by blow-up. In the second case, if \(\partial ^2_\mu S_\omega (Q^{\mu }_\omega )|_{\mu =1}\le 0\), we deduce that the ground state standing wave \(u(t,x)=e^{i\omega t}Q_\omega (x)\) is strongly unstable by blow-up, where \(S_\omega \) is the action, and \(Q_\omega ^{\mu }=\mu ^{3/2}Q_\omega (\mu x)\) is the \(L^2\)-invariant scaling.

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Acknowledgements

This work is supported by the National Natural Science Foundation of China (Nos. 11601435, 11801519).

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Authors and Affiliations

  1. Department of Mathematics, Northwest Normal University, Lanzhou, 730070, China

    Binhua Feng

  2. Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, China

    Qingxuan Wang

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  1. Binhua Feng
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  2. Qingxuan Wang
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Correspondence to Qingxuan Wang.

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Feng, B., Wang, Q. Strong Instability of Standing Waves for the Nonlinear Schrödinger Equation in Trapped Dipolar Quantum Gases. J Dyn Diff Equat 33, 1989–2008 (2021). https://doi.org/10.1007/s10884-020-09881-0

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  • DOI: https://doi.org/10.1007/s10884-020-09881-0

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