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.