Dynamics of Ginzburg-Landau vortices under a Landau-Lifshitz type flow

Motivated by the study of thin films of ferromagnetic
materials, we examine the motion of Ginzburg-Landau
vortices. The underlying equation is a hybrid of two
equations that are relatively well understood: a
nonlinear Schroedinger equation and a gradient flow.
But the two theories use different tools. One of them
relies on an analysis of the Jacobian of the solutions,
the other on the energy density. For the hybrid flow,
each of the methods yields a bad term that is difficult
to handle. However, if the two approaches are combined
in a suitable way, then we can take advantage of a
cancellation of the bad terms. We can then derive the
motion law for the vortices up to the first collision
time.

This is joint work with M. Kurzke (University of Bonn),
C. Melcher (University of Oxford), and D. Spirn
(University of Minnesota).