Solitons, solitonic vortices, and vortex rings in a confined Bose-Einstein condensate

Quasi-one-dimensional solitons that may occur in an elongated Bose-Einstein condensate become unstable at high particle density. We study two basic modes of instability and the corresponding bifurcations to genuinely three-dimensional solitary waves such as axisymmetric vortex rings and nonaxisymmetric solitonic vortices. We calculate the profiles of the above structures and examine their dependence on the velocity of propagation along a cylindrical trap. At sufficiently high velocity, both the vortex ring and the solitonic vortex transform into an axisymmetric soliton. We also calculate the energy-momentum dispersions and show that a Lieb-type mode appears in the excitation spectrum for all particle densities.