Multiple Time Scale Dynamics in Coupled Ginzburg-Landau Equations

Abstract

Using a rather simple model of coupled, time-dependent Ginzburg-Landau equations with two order parameters, we demonstrate that the total Hamilitonian energy of the system contains at least three levels describing point vortices, domain walls and configurations. The global in time dynamics contain then also at least three different time scales for nontrivial motions of domain walls, boundaries of domain walls (frational degree vortices) and paired vortices. In particular, we rigorously show, after an intial time period of adjusting, the domain walls start to move according to motion by the mean-curvature that straighten out the domain walls while the boundaries of such domain walls are essentially fixed. After this motion is completed, the fractional degree vortices begin to move at the next time scale. The motion is relatively simple as it is of constant speed and toward each other to form vortex pairs. Finally, these vortex pairs may move in the final time scale very much like the ordinary vortices in a single time-dependent Ginzburg-Laudau equation.