On the Bifurcation and Stability of Periodic Solutions of the Ginzburg--Landau Equations in the Plane

Abstract

The linear bifurcation and stability of periodic solutions to the Ginzburg--Landau equations in the plane are investigated. In particular, we find new infinite families of solutions, which include the few solutions previously reported in the literature. Then, the vortex structure of these new solutions is examined. In addition, the energy of a large class of solutions is approximated in the limit case for which the fundamental cell is a very thin and long rectangle. In that limit, we find that the energy of the solution representing the well-known triangular lattice is the lowest. Finally, we examine the stability of one infinite family of solutions, including both the triangular and square lattices, in an infinite-dimensional space of perturbations (in contrast to a previous work in which stability was examined only in a finite-dimensional space). We find that in addition to the triangular lattice other solutions are stable as well.