On the Gross–Pitaevskii equation for trapped dipolar quantum gases

Carles, Rémi; Markowich, Peter A.; Sparber, Christof

doi:10.1088/0951-7715/21/11/006

Cited by 89 publications

(136 citation statements)

References 26 publications

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“…Remark first that these two models can be considered in a unified way. For all u ∈ B 1 and for α ∈ {0, 1}, denote 12) where the convolution holds on the three variables (x, y, z) ∈ R 3 . Remark that for α = 0, this definition coincides with the definition (1.43).…”

Section: The Nonlinear Analysismentioning

confidence: 99%

“…The approximation of the Schrödinger-Poisson system with no magnetic field was studied when the electron gas is constraint in the vicinity of a plane in [7,25] and when the gas is constraint on a line in [5]. When the nonlinearity depends locally on the density, as for the Gross-Pitaevskii equation, asymptotic models for confined quantum systems were studied in [8,6,12]. In classical setting, collisional models in situations of strong confinement have been studied in [17].…”

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confidence: 99%

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An Effective Mass Theorem for the Bidimensional Electron Gas in a Strong Magnetic Field

Delebecque-Fendt

Méhats

2009

Commun. Math. Phys.

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Abstract. We study the limiting behavior of a singularly perturbed Schrödinger-Poisson system describing a 3-dimensional electron gas strongly confined in the vicinity of a plane (x, y) and subject to a strong uniform magnetic field in the plane of the gas. The coupled effects of the confinement and of the magnetic field induce fast oscillations in time that need to be averaged out. We obtain at the limit a system of 2-dimensional Schrödinger equations in the plane (x, y), coupled through an effective selfconsistent electrical potential. In the direction perpendicular to the magnetic field, the electron mass is modified by the field, as the result of an averaging of the cyclotron motion. The main tools of the analysis are the adaptation of the second order long-time averaging theory of ODEs to our PDEs context, and the use of a Sobolev scale adapted to the confinement operator.

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Section: The Nonlinear Analysismentioning

confidence: 99%

mentioning

confidence: 99%

An Effective Mass Theorem for the Bidimensional Electron Gas in a Strong Magnetic Field

Delebecque-Fendt

Méhats

2009

Commun. Math. Phys.

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“…For the study in mathematics, the existence and uniqueness as well as the possible blowup of solutions were studied in [12], and the existence of standing waves was proven in [13,14]. Due to the dipolar interaction potential of the nonlocal NLS, mathematical and numerical difficulties are introduced.…”

Section: Introductionmentioning

confidence: 99%

A mass conserved splitting method for the nonlinear Schrödinger equation

Hua

Zhu³

2012

Adv Differ Equ

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A mass conserved time-splitting difference method is presented for the onedimensional dipolar Bose-Einstein condensates (BECs) described by a nonlocal nonlinear Schrödinger equation with a convolution term. As a result of the singularity in the convolution term, it brings difficulties both in mathematical analysis and in numerical simulations. By properly using the difference scheme to deal with the convolution term, an imaginary time method is given to compute the ground states and then a time-splitting method is obtained for dynamics of dipolar BECs. This time-splitting numerical method is mass conserved everywhere, and it has second-order accuracy and is also unconditionally stable. Numerical results are given to verify the stability and energy conservation when there is no blow up. Mathematics Subject Classification (2010) 65M06; 35Q41.

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“…Yi and You [16] have firstly introduced a pseudo-potential appropriate to describe such systems. To our knowledge, the first rigorous mathematical study of ( [4]). An interesting question is to find new threshold criterion for (1.1), which guarantees the finite time blow up in the region λ 1 < 4π 3 λ 2 .…”

Section: Introductionmentioning

confidence: 99%

The threshold for the focusing Gross–Pitaevskii equation with trapped dipolar quantum gases

Cao

2011

Journal of Mathematical Analysis and Applications

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This paper concerns the threshold of global existence and finite time blow up of solutions to the time-dependent focusing Gross-Pitaevskii equation describing the BoseEinstein condensation of trapped dipolar quantum gases. Via a construction of new crossconstrained invariant sets, it is shown that either the corresponding solution globally exists or blows up in finite time according to some appropriate assumptions about the initial datum.

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On the Gross–Pitaevskii equation for trapped dipolar quantum gases

Cited by 89 publications

References 26 publications

An Effective Mass Theorem for the Bidimensional Electron Gas in a Strong Magnetic Field

An Effective Mass Theorem for the Bidimensional Electron Gas in a Strong Magnetic Field

A mass conserved splitting method for the nonlinear Schrödinger equation

The threshold for the focusing Gross–Pitaevskii equation with trapped dipolar quantum gases

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