Motions in a Bose condensate: IV. Axisymmetric solitary waves

Axisymmetric disturbances that preserve their form as they move through a Bose condensate are obtained numerically by the solution of the appropriate nonlinear Schrodinger equation. A continuous family is obtained that, in the momentum (p)-energy (E) plane, consists of two branches meeting at a cusp of minimum momentum around 0.140 PK~/C' and minimum energy about 0.145 p~~/c, where p is density, c is the speed of sound and K is the quantum of circulation. For all larger p, there are two possible energy states. One (the lower branch) is (for large enough p) a vortex ring of circulation K; as p+m its radius G-(~/TK)"* becomes infinite and its forward velocity tends to zero. The other (the upper branch) lacks vorticity and is a rarefaction sound pulse that becomes increasingly one dimensional as p +a; its velocity approaches c for large p. The velocity of any member of the family is shown, both numerically and analytically, to be aE/ap, the derivative being taken along the family. At great distances, the disturbance in the condensate is pseudo-dipolar (dipolar in a stretched coordinate system); the strength of the pseudo-dipole moment is obtained numerically. Analogous calculations are presen- ted for the corresponding two-dimensional problem. Again, a continuous sequence of solitary waves is obtained, but the momentum per unit length p and energy per unit length E have no minima. For small forward velocities, the wave consists of two widely separated parallel, oppositely directed line vortices. As the forward velocity increases the wave loses its vorticity and becomes a rarefaction pulse of ever increasing spatial extent but ever decreasing amplitude. As its velocity approaches c, both p and E tend to zero, and Elp + c.