Effective Dynamics of Magnetic Vortices

We study solutions of Ginzburg-Landau-type evolution equations (both dissipative and Hamiltonian) with initial data representing collections of widely-spaced vortices. We show that for long times, the solutions continue to describe collections of vortices, and we identify (to leading order in the vortex separation) the dynamical system describing the motion of the vortex centres (effective dynamics).