Vortex solutions of the evolutionary Ginzburg-Landau type equations

We consider two types of the time-dependent Ginzburg-Landau equation in 2D bounded domains: the heat-flow equation and the Schrödinger equation. We study the asymptotic behaviour of the vortex solutions of these equations when the vortex core size is much smaller than the inter-vortex distance. Using the method of the asymptotic expansion near the vortices, we obtain the systems of ordinary differential equations (ODEs) governing the evolution of the vortices. The expressions for these equations in the circle and in the annular domain are presented. We study the motion of the vortices in these two domains. It is shown that there exist the stationary points for both types of the equations and these points are determined for some particular cases. The heat-flow equation describes the particles that move like electric charges. If the vortices have different signs (i.e. Poincaré indices) then they attract each other and collide. If they have the same signs then they repulse. The particles always move away from the stationary points. The motion is very slow and non-periodic. The Schrödinger equation describes the motion of particles that behave like hydrodynamics vortices. The vortices with the same signs move in the same direction. If the signs are different then the vortices move in the opposite directions. In particular, if the initial positions are near the stationary points, then the particles move along the elliptic trajectories. The motion is not stable with respect to the initial data. It is always periodic or quasiperiodic. Examples of such trajectories are presented.