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.

We study the presence of exact localized solutions in a quadratic-cubic nonlinear Schrödinger equation with inhomogeneous nonlinearities. Using a specific ansatz, we transform the nonautonomous nonlinear equation into an autonomous one, which engenders composed states corresponding to solutions localized in space, with an oscillating behavior in time. Direct numerical simulations are employed to verify the stability of the modulated solutions against small random perturbations.

In this paper we study the effects of the transverse confinement strength on the dark solitons dynamics in quasi-one-dimensional Bose–Einstein condensates (BECs). The theoretical model is based on the Muoz-Mateo & Delgado (MMD) equation that describes cigar-shaped BECs with repulsive interatomic interactions. Since the MMD equation presents a nonpolynomial form, the soliton-sound recombination may not display the same pattern presented in the cubic model. We perform numerical simulations for harmonic and anharmonic (Gaussian) external potentials to compare both cases. All results via the MMD equation are in perfect agreement with the 3D simulations. Also, we noticed some differences between MMD and cubic models and changes in the soliton lifetime due to the values of the transverse confinement.