# American Institute of Mathematical Sciences

December  2011, 4(4): 831-856. doi: 10.3934/krm.2011.4.831

## 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, France 2 Department of Mathematics, National University of Singapore, Singapore 119076, Singapore 3 IRMAR, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France, France

Received  May 2011 Revised  September 2011 Published  November 2011

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.
Citation: Naoufel Ben Abdallah, Yongyong Cai, Francois Castella, Florian Méhats. Second order averaging for the nonlinear Schrödinger equation with strongly anisotropic potential. Kinetic & Related Models, 2011, 4 (4) : 831-856. doi: 10.3934/krm.2011.4.831
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