The dynamics of line singularities in 3 different systems were considered. These were: vortices in inviscid fluids, vortices in type-II superconductors and dislocations in elastic crystals. It was noted that, when the core of the singularity could be regularized within a continuum model, as in the case of superconducting and fluid vortices, the dynamics could be derived systematically in the asymptotic limit as the core radius tended to zero. Such an asymptotic analysis was more difficult when the core of the singularity was so small that it required atomistic modelling, as in the case of dislocations. In each situation, the derivation of a law of motion for the singularity was approached by using the same basic methodology. The outer problem was linear, but singular, and the variables of interest diverged at the singularity. This led to an inner problem near to the core of the singularity, over which some regularizing mechanism acted. The leading order (in core radius) inner problem was 2-dimensional when the line which described the singularity in the outer problem was smooth. It was often radially symmetrical as well. At the next order in the inner solution, the Fredholm alternative could be used to derive a solvability condition for the first-order equations. When this inner solution was matched to the outer solution, this solvability condition gave the law of motion of the singularity. This procedure was successfully carried out for fluid vortices and superconducting vortices, in a variety of situations.

Dynamics of Line Singularities. A.Carpio, S.J.Chapman, S.D.Howison, J.R.Ockendon: Philosophical Transactions of the Royal Society, 1997, 355, 2013-24