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

Yongming Luo; Athanasios Stylianou

doi:10.3934/dcdsb.2020239

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2021, Volume 26, Issue 6: 3455-3477. Doi: 10.3934/dcdsb.2020239

This issue Previous Article Dynamic analysis on an almost periodic predator-prey system with impulsive effects and time delays Next Article

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

Yongming Luo^1, , and
Athanasios Stylianou^2,

1.
Institut für Wissenschaftliches Rechnen, Technische Universität Dresden, Willers-Bau, Zellescher Weg 12-14, 01069 Dresden, Germany
2.
Institut für Mathematik, Universität Kassel, Heinrich-Plett-Straße 40, 34132 Kassel, Germany

^* Corresponding author: Yongming Luo
^* Corresponding author: Yongming Luo

Received: April 30, 2020

Revised: May 31, 2020

Early access: August 2020

Published: June 2021

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Abstract

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.

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

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