Motions in a Bose condensate. IV. Axisymmetric solitary waves

For pt.III see ibid., vol.7, p.260 (1974). 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 rho kappa 3/c2 and minimum energy about 0.145 rho kappa 3/c, where rho is density, c is the speed of sound and kappa 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 kappa ; as p to infinity its radius omega approximately (p/ pi kappa )1/2 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 to infinity ; its velocity approaches c for large p. The velocity of any member of the family is shown, both numerically and analytically, to be delta E/ delta p, the derivative being taken along the family.