Deviation from one dimensionality in stationary properties and collisional dynamics of matter-wave solitons

By means of analytical and numerical methods, we study how the residual three dimensionality affects dynamics of solitons in an attractive Bose-Einstein condensate loaded into a cigar-shaped trap. Based on an effective one-dimensional (1D) Gross-Pitaevskii equation that includes an additional quintic self-focusing term, generated by the tight transverse confinement, we find a family of exact one-soliton solutions and demonstrate stability of the entire family, despite the possibility of collapse in the 1D equation with the quintic self-focusing nonlinearity. Simulating collisions between two solitons in the same setting, we find a critical velocity, Vc, below which merger of identical in-phase solitons is observed. Dependence of Vc on the strength of the transverse confinement and number of atoms in the solitons is predicted by means of the perturbation theory and investigated in direct simulations. The simulations also demonstrate symmetry breaking in collisions of identical solitons with a nonzero phase difference. This effect is qualitatively explained by means of an analytical approximation.