arXiv:0805.0716 · Nonlinearity

On the Gross–Pitaevskii equation for trapped dipolar quantum gases

Rémi Carles, Peter A. Markowich, Christof Sparber

Published Sep 29, 2008• 79 citations

Abstract

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.

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

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arXiv:0805.0716 · Nonlinearity

On the Gross–Pitaevskii equation for trapped dipolar quantum gases

Rémi Carles, Peter A. Markowich, Christof Sparber

Published Sep 29, 2008• 79 citations

arXiv HTML PDF

Abstract

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

Related papers

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Peter A. Markowich, Christof Sparber

Jun 21, 201582 citations

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Journal of Mathematical PhysicsDec 1, 20195 citations

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Weizhu Bao, Naoufel Ben Abdallah, Yongyong Cai

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(-\Delta)^{1/2}

is involved which brings significant difficulties into the analysis. Existence and uniqueness as well as nonexistence of the ground state under different parameter regimes are established for the quasi-2D and quasi-1D equations. Well-posedness of the Cauchy problem for both types of equations and finite time blow-up in two dimensions are analyzed. Finally, we rigorously prove the convergence with linear convergence rate for the solutions of the 3D GPPS and its quasi-2D and quasi-1D approximate equations in the weak interaction regime.

\varepsilon

is a small positive parameter,

\lambda_1, \lambda_2\in \mathbb{R}

*

denotes the convolution,

K(x) = \frac{{1 - 3{{\cos }^2}\theta }}{{{{| x |}^3}}}

and

\theta = \theta(x)

is the angle between the dipole axis determined by

(0,0,1)

and the vector

x

. Moreover, the potential

V

satisfies

\liminf_{|x| \to \infty } V(x) > \inf_{\mathbb{R}^3} V(x) = 0

. Under certain assumptions on

(\lambda_1, \lambda_2)\in \mathbb{R}^2

, we construct a family of positive solutions

{u_\varepsilon } \in {H^1}({\mathbb{R}^3})

whose

{L^\infty }

norm approaches

0

\varepsilon \to 0

. Our main results extend the results in Byeon and Wang [6] which dealt with singularly perturbed Schrodinger equations with a local nonlinearity, to the nonlocal Gross-Pitaevskii type equation. For more information and the latex file, see https://ejde.math.txstate.edu/Volumes/2026/48/abstr.html