Approach and separation of quantum vortices with balanced cores

Using two innovations, smooth but different, scaling laws for the reconnection of pairs of initially orthogonal and anti-parallel quantum vortices are obtained using the three-dimensional Gross-Pitaevskii equations. For the anti-parallel case, the scaling laws just before and after reconnection obey the dimensional $\delta\sim|t-t_r|^{1/2}$ prediction with temporal symmetry about the reconnection time $t_r$ and physical space symmetry about $x_r$, the mid-point between the vortices, with extensions forming the edges of an equilateral pyramid. For all of the orthogonal cases, before reconnection $\delta_{in}\sim(t-t_r)^{1/3}$ and after reconnection $\delta_{out}\sim(t_r-t)^{2/3}$, which are respectively slower and faster than the dimensional prediction. In these cases, the reconnection takes place in a plane defined by the directions of the curvature and vorticity.