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

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. •A new effective unidimensional Nonlinear Schroedinger Equation is proposed.•The procedure of dimensional reduction can be applied in another confinements.•The nonlinearity is written in terms of the Lambert W function.•Comparison between the anti-Gaussian and the Harmonic potentials is made.