In the Ginzburg-Landau model for superconductivity a large Ginzburg-Landau parameter κ corresponds to the formation of tight, stable vortices. These vortices are located where an applied magnetic field pierces the superconducting bulk, and each vortex induces a quantized supercurrent about the vortex. The energy of large-κ solutions blows up near each vortex which brings about difficulties in a...

We consider 2D localized rotating patterns which solve a parabolic system of PDEs on the spatial domain R2. Under suitable assumptions, we prove nonlinear stability with asymptotic phase with respect to the norm in the Sobolev space H2. The stability result is obtained by a combination of energy and resolvent estimates, after the dynamics is decomposed into an evolution within a three–dimension...

In this paper, we will construct the solution of Landau-Ginzburg equation by Adomian decomposition method. This method avoids linearization space and discretization time, it often gives a good approximation exact solution.

We continue the work on backstepping controllers previously designed for the Ginzburg-Landau model of vortex shedding developed for the 2D case by adding another dimension. In the present paper we show that the control and observer designs for the 2D case extend to the 3D case, and demonstrate for the first time their effectiveness in attenuating vortex shedding in GL simulations.