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

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.