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.

中文翻译：

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雪茄形玻色-爱因斯坦凝聚态的 Lambert W 函数非线性的有效一维方程


在本文中，我们根据 Mateo 和 Delgado 的工作 [Phys. Rev. A 77， 013617 （2008）] 推导出了一个有效的一维方程，该方程控制平均场雪茄形玻色-爱因斯坦凝聚态 （BEC） 的轴向动力学，具有排斥性的原子间相互作用，并受横向反高斯限制电位的影响。得到的方程表现出非标准非线性，用 Lambert W 函数的主分支写成。该方程适当地包含了 Thomas-Fermi （TF） 和微扰状态中的正确极限。我们的研究结果的有效性是通过与从完整的三维 Gross-Pitaevskii 方程得出的数值解进行比较分析来确定的，包括关于 TF 和扰动机制中遇到的限制情况的解。