Singular configurations known as topological defects are found in the ordered phases of numerous two dimensional  systems, in and out of equilibrium, from liquid crystals to superconductors.

Their dynamics is modeled as a particle-field problem, with the orientation field acting as the interaction potential produced by particle-like singularities.

While most of the studies focus on the study of quasi-static, isotropic defects way fewer theoretical results have been obtained outside of this regime.  To address this issue, we solve the multivalued diffusion equation for the orientation field of a defect with an arbitrary trajectory, revealing qualitatively new features that emerge as a result of the system's memory on the previous position of the singularity. We then propose a novel framework for the derivation of the equation ruling the motion of the defects and use it to compute the trajectories of annihilating pairs in liquid crystals with elastic anisotropy.  Our results accord well with direct simulations of the full-order parameter dynamics.