On 3d dipolar BoseEinstein condensates involving quantum fluctuations and threebody interactions
Abstract
We study the following nonlocal mixed order GrossPitaevskii equation $$i\,\partial_t \psi=\frac{1}{2}\,\Delta \psi+V_{ext}\,\psi+\lambda_1\,\psi^2\,\psi+\lambda_2\,(K*\psi^2)\,\psi+\lambda_3\,\psi^{p2}\,\psi,$$ where $K$ is the classical dipoledipole 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 stateoftheart model for describing the dynamics of dipolar BoseEinstein condensates (LeeHuangYang corrected dipolar GPE). We prove existence and nonexistence of standing waves in different parameter regimes; for $p\neq 6$ we prove global wellposedness and small data scattering.
 Publication:

arXiv eprints
 Pub Date:
 February 2019
 DOI:
 10.48550/arXiv.1902.05591
 arXiv:
 arXiv:1902.05591
 Bibcode:
 2019arXiv190205591L
 Keywords:

 Mathematics  Analysis of PDEs;
 35Q55;
 49J35;
 35B09