Generalized one-dimensional nonpolynomial Schr\"odinger equation for Bose-Einstein condensates with generic transverse confinement

TL;DR

This paper derives a generalized one-dimensional nonpolynomial Schr"odinger equation for Bose-Einstein condensates with arbitrary transverse confinement, validated by numerical tests showing excellent agreement with full 3D simulations for both static and dynamic properties.

Contribution

It introduces a new effective 1D model for BECs with generic transverse potentials, extending previous models to more general geometries and confirming its accuracy through comprehensive numerical validation.

Findings

Accurately reproduces ground-state properties across interaction regimes.

Successfully models matter-wave scattering and transmission.

Demonstrates robustness of the generalized equation in dynamic simulations.

Abstract

This work presents a dimensional reduction of Bose-Einstein condensates confined by generalized transverse potentials, parametrized by an exponent $n$. Starting from the three-dimensional Gross-Pitaevskii equation, we employ a variational ansatz to derive an effective one-dimensional nonpolynomial Schr\"odinger equation, which self-consistently determines the transverse width dynamics. The model generalizes existing formalisms for cigar- and funnel-shaped geometries. We validate the approach through comprehensive numerical tests, demonstrating excellent agreement with full 3D simulations for ground-state properties across various interaction regimes. Finally, real-time simulations of matter-wave scattering at potential barriers verify the model's dynamical robustness, successfully replicating the spatiotemporal evolution and energy-dependent transmission characteristics observed in full 3D calculations.