  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
non-axisymmetric 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.
