Quasi-one-dimensional solitons that may occur in an elongated Bose-Einsteincondensate become unstable at high particle density. We study two basic modesof instability and the corresponding bifurcations to genuinelythree-dimensional solitary waves such as axisymmetric vortex rings andnon-axisymmetric solitonic vortices. We calculate the profiles of the abovestructures and examine their dependence on the velocity of propagation along acylindrical trap. At sufficiently high velocity, both the vortex ring and thesolitonic vortex transform into an axisymmetric soliton. We also calculate theenergy-momentum dispersions and show that a Lieb-type mode appears in theexcitation spectrum for all particle densities.
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