Second order averaging for the nonlinear Schrödinger equation with strongly anisotropic potential

Naoufel Ben Abdallah; Yongyong Cai; Francois Castella; Florian Méhats

doi:10.3934/krm.2011.4.831

Article Contents

2011, Volume 4, Issue 4: 831-856. Doi: 10.3934/krm.2011.4.831

This issue Previous Article Preface Next Article On the minimization problem of sub-linear convex functionals

Second order averaging for the nonlinear Schrödinger equation with strongly anisotropic potential

1.
IMT, UMR CNRS 5219, Université Paul Sabatier, 118, route de Narbonne, 31062 Toulouse Cedex
2.
Department of Mathematics, National University of Singapore, Singapore 119076
3.
IRMAR, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex

Received: April 30, 2011

Revised: August 31, 2011

Published: November 2011

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Abstract

We consider the three dimensional Gross-Pitaevskii equation\break (GPE) describing a Bose-Einstein Condensate (BEC) which is highly confined in vertical $z$ direction. The confining potential induces high oscillations in time. If the confinement in the $z$ direction is a harmonic trap -- an approximation which is widely used in physical experiments -- the very special structure of the spectrum of the confinement operator implies that the oscillations are periodic in time. Based on this observation, it can be proved that the GPE can be averaged out with an error of order of $\epsilon$, which is the typical period of the oscillations. In this article, we construct a more accurate averaged model, which approximates the GPE up to errors of order $\mathcal{O}(\epsilon^2)$. Then, expansions of this model over the eigenfunctions (modes) of the confining operator $H_z$ in the $z$-direction are given in view of numerical applications. Efficient numerical methods are constructed to solve the GPE with cylindrical symmetry in 3D and the approximation model with radial symmetry in 2D, and numerical results are presented for various kinds of initial data.

Keywords:

Gross-Pitaevskii equation,
dimension reduction,
second order averaging,
Laguerre-Hermite pseudo-spectral method.

Mathematics Subject Classification: 34C29, 35Q55, 46E35, 65M70.

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