# American Institute of Mathematical Sciences

June  2021, 26(6): 3455-3477. doi: 10.3934/dcdsb.2020239

## 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  June 2021 Early access  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, 2021, 26 (6) : 3455-3477. doi: 10.3934/dcdsb.2020239
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##### References:
 [1] Xiaoyu Zeng, Yimin Zhang. Asymptotic behaviors of ground states for a modified Gross-Pitaevskii equation. Discrete & Continuous Dynamical Systems, 2019, 39 (9) : 5263-5273. doi: 10.3934/dcds.2019214 [2] Patrick Henning, Johan Wärnegård. Numerical comparison of mass-conservative schemes for the Gross-Pitaevskii equation. Kinetic & Related Models, 2019, 12 (6) : 1247-1271. doi: 10.3934/krm.2019048 [3] Ricardo A. Pastrán, Oscar G. Riaño. Sharp well-posedness for the Chen-Lee equation. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2179-2202. doi: 10.3934/cpaa.2016033 [4] Georgy L. Alfimov, Pavel P. Kizin, Dmitry A. Zezyulin. Gap solitons for the repulsive Gross-Pitaevskii equation with periodic potential: Coding and method for computation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1207-1229. doi: 10.3934/dcdsb.2017059 [5] Roy H. Goodman, Jeremy L. Marzuola, Michael I. Weinstein. Self-trapping and Josephson tunneling solutions to the nonlinear Schrödinger / Gross-Pitaevskii equation. Discrete & Continuous Dynamical Systems, 2015, 35 (1) : 225-246. doi: 10.3934/dcds.2015.35.225 [6] Norman E. Dancer. On the converse problem for the Gross-Pitaevskii equations with a large parameter. Discrete & Continuous Dynamical Systems, 2014, 34 (6) : 2481-2493. doi: 10.3934/dcds.2014.34.2481 [7] E. Norman Dancer. On a degree associated with the Gross-Pitaevskii system with a large parameter. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 1835-1839. doi: 10.3934/dcdss.2019120 [8] Philipp Bader, Sergio Blanes, Fernando Casas, Mechthild Thalhammer. Efficient time integration methods for Gross-Pitaevskii equations with rotation term. Journal of Computational Dynamics, 2019, 6 (2) : 147-169. doi: 10.3934/jcd.2019008 [9] Thomas Chen, Nataša Pavlović. On the Cauchy problem for focusing and defocusing Gross-Pitaevskii hierarchies. Discrete & Continuous Dynamical Systems, 2010, 27 (2) : 715-739. doi: 10.3934/dcds.2010.27.715 [10] Ko-Shin Chen, Peter Sternberg. Dynamics of Ginzburg-Landau and Gross-Pitaevskii vortices on manifolds. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 1905-1931. doi: 10.3934/dcds.2014.34.1905 [11] Hyungjin Huh, Bora Moon. Low regularity well-posedness for Gross-Neveu equations. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1903-1913. doi: 10.3934/cpaa.2015.14.1903 [12] Dong Deng, Ruikuan Liu. Bifurcation solutions of Gross-Pitaevskii equations for spin-1 Bose-Einstein condensates. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3175-3193. doi: 10.3934/dcdsb.2018306 [13] Yujin Guo, Xiaoyu Zeng, Huan-Song Zhou. Blow-up solutions for two coupled Gross-Pitaevskii equations with attractive interactions. Discrete & Continuous Dynamical Systems, 2017, 37 (7) : 3749-3786. doi: 10.3934/dcds.2017159 [14] Jeremy L. Marzuola, Michael I. Weinstein. Long time dynamics near the symmetry breaking bifurcation for nonlinear Schrödinger/Gross-Pitaevskii equations. Discrete & Continuous Dynamical Systems, 2010, 28 (4) : 1505-1554. doi: 10.3934/dcds.2010.28.1505 [15] Shuai Li, Jingjing Yan, Xincai Zhu. Constraint minimizers of perturbed gross-pitaevskii energy functionals in $\mathbb{R}^N$. Communications on Pure & Applied Analysis, 2019, 18 (1) : 65-81. doi: 10.3934/cpaa.2019005 [16] Shouming Zhou, Chunlai Mu, Liangchen Wang. Well-posedness, blow-up phenomena and global existence for the generalized $b$-equation with higher-order nonlinearities and weak dissipation. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 843-867. doi: 10.3934/dcds.2014.34.843 [17] Boris Kolev. Local well-posedness of the EPDiff equation: A survey. Journal of Geometric Mechanics, 2017, 9 (2) : 167-189. doi: 10.3934/jgm.2017007 [18] Lin Shen, Shu Wang, Yongxin Wang. The well-posedness and regularity of a rotating blades equation. Electronic Research Archive, 2020, 28 (2) : 691-719. doi: 10.3934/era.2020036 [19] Jerry Bona, Nikolay Tzvetkov. Sharp well-posedness results for the BBM equation. Discrete & Continuous Dynamical Systems, 2009, 23 (4) : 1241-1252. doi: 10.3934/dcds.2009.23.1241 [20] A. Alexandrou Himonas, Curtis Holliman. On well-posedness of the Degasperis-Procesi equation. Discrete & Continuous Dynamical Systems, 2011, 31 (2) : 469-488. doi: 10.3934/dcds.2011.31.469

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