Effective equation for quasi-one dimensional tube-shaped Bose–Einstein condensates

Abstract

In this letter we derive an effective 1D equation that describes the axial dynamics of a tube-shaped Bose–Einstein condensate. The dimensional reduction is achieved by using a variational approach starting from the 3D Gross–Pitaevskii equation (GPE) in presence of a nonharmonic external potential in the transverse direction and generic in the axial one. The resulting equation is a time-dependent 1D nonpolynomial Schrödinger equation (NPSE). In view to check the accuracy of our 1D-NPSE, we numerically investigated the ground state properties of such a system that are in perfect agreement with the results produced by the 3D-GPE. We also compare the results with those from an 1D cubic nonlinear Schrödinger equation and the Thomas–Fermi approximation. Finally, the dynamics of ground states obtained from our 1D-NPSE is verified numerically by considering a small change in the strength of the axial confining potential.