Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

ZAMM ‐ Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik

Blow up threshold for the Gross‐Pitaevskii system with trapped dipolar quantum gases

Baiyu Liu, Li Ma, Jing Wang

Published Apr 2, 2015• 2 citations
Publisher

Abstract

In this paper, we study the Gross‐Pitaevskii system with trapped dipolar quantum gases. We obtain both the stable regime and the unstable regime. Moreover, via a construction of cross minimization problem, the blow up threshold is established.
DOI
10.1002/zamm.201400189

Related papers

Blow-up for trapped dipolar quantum gases with large energy

Yanfang Gao, Zhiyong Wang

We consider a Gross–Pitaevskii equation with harmonic potential arising in dipolar Bose–Einstein condensate. We extend the scope of blow-up solutions in the work of Carles, Markowich, and Sparber [Nonlinearity 21(11), 2569–2590 (2008)] to those with arbitrarily large energy in a partial unstable regime.

Journal of Mathematical Physics•Dec 1, 2019•5 citations

On the Gross–Pitaevskii equation for trapped dipolar quantum gases

Rémi Carles, Peter A. Markowich, Christof Sparber

We study the time-dependent Gross–Pitaevskii equation describing Bose–Einstein condensation of trapped dipolar quantum gases. Existence and uniqueness as well as the possible blow-up of solutions are studied. Moreover, we discuss the problem of dimension reduction for this nonlinear and nonlocal Schrödinger equation.

arXiv:0805.0716•Nonlinearity•Sep 29, 2008•79 citations

8 ON THE GROSS–PITAEVSKII EQUATION FOR TRAPPED DIPOLAR QUANTUM GASES

Peter A. Markowich, Christof Sparber

We study the time-dependent Gross–Pitaevskii equation describing Bose–Einstein condensation of trapped dipolar quantum gases. Existence and uniqueness as well as the possible blow-up of solutions are studied. Moreover, we discuss the problem of dimension reduction for this nonlinear and nonlocal Schrödinger equation.

•Jun 21, 2015•82 citations

Mass-energy threshold dynamics for dipolar quantum gases

Van Duong Dinh, Luigi Forcella, Hichem Hajaiej

We consider a Gross-Pitaevskii equation which appears as a model in the description of dipolar Bose-Einstein condensates, without a confining external trapping potential. We describe the asymptotic dynamics of solutions to the corresponding Cauchy problem in the energy space in different configurations with respect to the mass-energy threshold, namely for initial data above and at the mass-energy threshold. We first establish a scattering criterion for the equation that we prove by means of the concentration/compactness and rigidity scheme. This criterion enables us to show the energy scattering for solutions with data above the mass-energy threshold, for which only blow-up was known. We also prove a blow-up/grow-up criterion for the equation with general data in the energy space. As a byproduct of scattering and blow-up criteria, and the compactness of minimizing sequences for the Gagliardo-Nirenberg's inequality, we study long time dynamics of solutions with data lying exactly at the mass-energy threshold.

arXiv:2009.05933•Communications in Mathematical Sciences•Dec 10, 2021•11 citations

Going beyond the threshold: Blowup criteria with arbitrary large energy in trapped quantum gases

Lingfei Li, Yingying Xie, Yongsheng Yan +1 more

The present paper considers the blowup properties in trapped dipolar quantum gases modelled by the Gross-Pitaevskii equation. More precisely, through analyzing the temporal evolution of $ J'(t) $ in the form of uncertain principle [ 1 ], we provide some invariant evolution flows. Based on that, we establish the global existence versus blowup dichotomy of solutions above the mass-energy threshold. Meanwhile, we can estimate the behaviour of solutions with arbitrary large energy.

AIMS Mathematics•Jan 1, 2022•0 citations

Complementary study of the standing wave solutions of the Gross-Pitaevskii equation in dipolar quantum gases

Rémi Carles, Hichem Hajaiej

We study the stability of the standing wave solutions of a Gross–Pitaevskii equation describing Bose–Einstein condensation of dipolar quantum gases and characterize their orbit. As an intermediate step, we consider the corresponding constrained minimization problem and establish existence, symmetry and uniqueness of the ground state solutions.

arXiv:1402.7267•Bulletin of the London Mathematical Society•Apr 26, 2015•30 citations
We consider a Gross-Pitaevskii equation which appears as a model in the description of dipolar Bose-Einstein condensates, without a confining external trapping potential. We describe the asymptotic dynamics of solutions to the corresponding Cauchy problem in the energy space in different configurations with respect to the mass-energy threshold, namely for initial data above and at the mass-energy threshold. We first establish a scattering criterion for the equation that we prove by means of the concentration/compactness and rigidity scheme. This criterion enables us to show the energy scattering for solutions with data above the mass-energy threshold, for which only blow-up was known. We also prove a blow-up/grow-up criterion for the equation with general data in the energy space. As a byproduct of scattering and blow-up criteria, and the compactness of minimizing sequences for the Gagliardo-Nirenberg's inequality, we study long time dynamics of solutions with data lying exactly at the mass-energy threshold.