Effective Equations for Repulsive Quasi‐One Dimensional Bose–Einstein Condensates Trapped with Anharmonic Transverse Potentials

One‐dimensional nonlinear Schrödinger equations are derived to describe the axial effective dynamics of cigar‐shaped atomic repulsive Bose‐Einstein condensates trapped with anharmonic transverse potentials. The accuracy of these equations in the perturbative, Thomas‐Fermi, and crossover regimes were verified numerically by comparing the ground‐state profiles, transverse chemical potentials and oscillation patterns with those results obtained for the full three‐dimensional Gross‐Pitaevskii equation. This procedure allows us to derive different patterns of 1D nonlinear models by the control of the transverse confinement even in the presence of an axial vorticity. The longitudinal dynamics of a cigar‐shaped monoatomic Bose‐Einstein condensate transversally trapped by an anharmonic potential can be accurately described by the 1D perturbative and Thomas‐Fermi models valid in the perturbative interaction regime and in the strong interaction regime, respectively. In the crossover regime, there is not an effective model up to now. A new effective model made to fill this gap is developed in this paper.