A simplified model for the energy of the magnetization of a thin ferromagnetic film gives rise to a version of the theory of GinzburgLandau vortices for sphere-valued maps. In particular we have the development of vortices as a certain parameter tends to 0. The dynamics of the magnetization is ruled by the Landau-Lifshitz-Gilbert equation, which combines characteristic properties of a nonlinear...

We consider the Ginzburg-Landau equation, @ t u = @ 2 x u+u?ujuj 2 , with complex amplitude u(x; t). We first analyze the phenomenon of phase slips as a consequence of the local shape of u. We next prove a global theorem about evolution from an Eckhaus unstable state, all the way to the limiting stable finite state, for periodic perturbations of Eckhaus unstable periodic initial data. Equipped ...