  The Gross-Pitaevskii (GP) equation admits a two-dimensional solitary wave
solution representing two mutually self-propelled, anti-parallel straight line
vortices. The complete sequence of such solitary wave solutions has been
computed by Jones and Roberts (J. Phys. A, 15, 2599, 1982). These solutions are
unstable with respect to three-dimensional perturbations (the Crow
instability). The most unstable mode has a wavelength along the direction of
the vortices of the same order as their separation. The growth rate associated
with this mode is evaluated here, and it is found to increase very rapidly with
decreasing separation. It is shown, through numerical integrations of the GP
equation that, as the perturbations grow to finite amplitude, the lines
reconnect to produce a sequence of almost circular vortex rings.
