Effective wave equations for the dynamics of cigar-shaped and disk-shaped Bose condensates

TLDR

The trapping potential is harmonic transversely and generic axially, and the derived 1D NPSE models essentially one‑dimensional dynamics of cigar‑shaped Bose condensates. The study derives an effective one‑dimensional wave equation for the axial dynamics of a Bose condensate in a cylindrically symmetric external potential. Using a variational approach to the 3D Gross‑Pitaevskii equation, the authors obtain a 1D NPSE for cigar‑shaped condensates and a 2D NPSE for disk‑shaped condensates. The 1D NPSE yields far more accurate predictions than previous models, provides analytical bright and dark soliton solutions, and reduces to Schrödinger‑like equations with varying nonlinearities in weak and strong interaction limits.

Abstract

Starting from the three-dimensional (3D) Gross-Pitaevskii equation and using a variational approach, we derive an effective 1D wave equation that describes the axial dynamics of a Bose condensate confined in an external potential with cylindrical symmetry. The trapping potential is harmonic in the transverse direction and generic in the axial one. Our equation, that is a time-dependent nonpolynomial nonlinear Schr\"odinger equation (1D NPSE), can be used to model cigar-shaped condensates, whose dynamics is essentially 1D. We show that 1D NPSE gives much more accurate results than all other effective equations recently proposed. By using 1D NPSE we find analytical solutions for bright and dark solitons, which generalize the ones known in the literature. We deduce also an effective 2D nonpolynomial Schr\"odinger equation (2D NPSE) that models disk-shaped Bose condensates confined in an external trap that is harmonic along the axial direction and generic in the transverse direction. In the limiting cases of weak and strong interaction, our approach gives rise to Schr\"odinger-like equations with different polynomial nonlinearities.