Abstract
We consider the three dimensional Gross-Pitaevskii equation (GPE) describing a Bose-Einstein Condensate (BEC) which is highly confi ned in vertical z direction. The highly confi ned potential induces high oscillations in time. If the confi nement in the z direction is a harmonic trap (which is widely used in physical experiments), the very special structure of the spectrum of the confi nement operator will imply 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 epsilon squared. Then, expansions of this model over the eigenfunctions (modes) of the vertical Hamiltonian Hz are given in convenience of numerical application. Effi cient numerical methods are constructed for solving 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.
Highlights
In this paper, we consider the approximation of the Gross-Pitaevskii equation (GPE) modeling the evolution of anisotropic Bose-Einstein Condensates in the space (x, z) ∈ R2 × R
For partially confined Bose-Einstein Condensate (BEC) (??), the early works [?, ?] focused on the initial data concentrated on the ground mode of Hz
We focus on the three Cauchy problems, the initial problem (??), the averaged problem (??) and the second order approximation (??)
Summary
We consider the approximation of the Gross-Pitaevskii equation (GPE) modeling the evolution of anisotropic Bose-Einstein Condensates in the space (x, z) ∈ R2 × R. For partially confined BEC (??), the early works [?, ?] focused on the initial data concentrated on the ground mode of Hz. For general initial data case, following the ideas in [?, ?], it is desirable to filter out the oscillations and to derive approximate models for (??) by using averaging techniques. In [?], Ben Abdallah et al have developed the averaging technique and proved that, for general confining potentials in the z direction, the limiting model as ε goes to zero is i∂tΦ = HxΦ + Fav (Φ) , Φ(t = 0) = Ψinit,. The important point is that Hz admits only integers as eigenvalues, the function F is 2π-periodic with respect to the τ variable In such case, the result of [?] can be specified.
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