E. Barany, M. Golubitsky and J. Turski
Bifurcations with local gauge symmetries in the Ginzburg-Landau equations
Physica D. 56 (1992) 36-56.
An interesting class of physical systems are those that exhibit local gauge symmetries: internal invariances that can be implemented independently at any space-time point. Systems in which these symmetries are spontaneously broken exhibit remarkable properties such as superconductivity, and if such systems also possess spatial symmetry, pattern formation can accompany the gauge symmetry-breaking. We conduct a careful analysis of a well-known example of this phenomenon: the formation of the Abrikosov vortex lattice in the Ginzburg-Landau model of Type-II superconductors. The study of this system has a long history and our principal contribution is to put the analysis rigorously into the context of steady-state equivariant bifurcation theory by the proper implementation of a gauge-fixing procedure. This example may be typical of the way that gauge and spatial symmetries intertwine to produce spatial patterns.