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On 3d dipolar Bose-Einstein condensates involving quantum fluctuations and three-body interactions

  • * Corresponding author: Yongming Luo

    * Corresponding author: Yongming Luo 
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  • We study the following nonlocal mixed order Gross-Pitaevskii equation

    $ i\,\partial_t \psi = -\frac{1}{2}\,\Delta \psi+V_{ext}\,\psi+\lambda_1\,|\psi|^2\,\psi+\lambda_2\,(K*|\psi|^2)\,\psi+\lambda_3\,|\psi|^{p-2}\,\psi, $

    where $ K $ is the classical dipole-dipole interaction kernel, $ \lambda_3>0 $ and $ p\in(4,6] $; the case $ p = 6 $ being energy critical. For $ p = 5 $ the equation is considered currently as the state-of-the-art model for describing the dynamics of dipolar Bose-Einstein condensates (Lee-Huang-Yang corrected dipolar GPE). We prove existence and nonexistence of standing waves in different parameter regimes; for $ p\neq 6 $ we prove global well-posedness and small data scattering.

    Mathematics Subject Classification: Primary: 35Q55, 49J35; Secondary: 35B09.


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