# American Institute of Mathematical Sciences

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

 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

Received  May 2020 Revised  June 2020 Published  August 2020

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.
Citation: Yongming Luo, Athanasios Stylianou. On 3d dipolar Bose-Einstein condensates involving quantum fluctuations and three-body interactions. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020239
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