  Axisymmetric disturbances that preserve their form as they move along the
vortex lines in uniform Bose-Einstein condensates are obtained numerically by
the solution of the Gross-Pitaevskii equation. A continuous family of such
solitary waves is shown in the momentum ($p$) -- substitution energy
($\hat{\cal E}$) plane with $p\to 0.09 \rho \kappa^3/c^2, \hat{\cal E}\to 0.091
\rho \kappa^3/c$ as $U \to c$, where $\rho$ is the density, $c$ is the speed of
sound, $\kappa$ is the quantum of circulation and $U$ is the solitary wave
velocity. It is shown that collapse of a bubble captured by a vortex line leads
to the generation of such solitary waves in condensates. The various stages of
collapse are elucidated. In particularly, it is shown that during collapse the
vortex core becomes significantly compressed and after collapse two solitary
wave trains moving in opposite directions are formed on the vortex line.
