Effective 1D-equation with a Lambert W function nonlinearity for cigar-shaped Bose–Einstein condensates

Abstract

In this paper, we derive an effective 1D equation based on the work of Mateo and Delgado [Phys. Rev. A 77, 013617 (2008)] that governs the axial dynamics of mean-field cigar-shaped Bose–Einstein condensates (BECs) with repulsive interatomic interactions and subject to transverse anti-Gaussian confining potential. The resulting equation exhibits a nonstandard nonlinearity written in terms of the principal branch of the Lambert W function. This equation appropriately encompasses the correct limits within the Thomas–Fermi (TF) and perturbative regimes. The validity of our findings was established through a comparative analysis with numerical solutions derived from the complete three-dimensional Gross–Pitaevskii equation, including those about the limiting cases encountered in the TF and perturbative regimes.