Vortex interaction and the two-dimensional dark soliton

We study vortex interaction and a two-dimensional dark soliton in the two-dimensional Gross-Pitaevskii equation. A vortex-antivortex pair propagates with a constant velocity when the initial distance is large. But complicated behaviors appear when the initial distance of two vortices becomes smaller. If the initial distance is sufficiently small, a two-dimensional dark solution is created as a result of the pair annihilation. The two-dimensional dark soliton has an anisotropic structure and propagates in a certain direction. The two-dimensional dark soliton is stable against head-on collision.