The Complex Ginzburg-Landau equation for beginners

Several systems discussed at this workshop on Spatio-Temporal Patterns in Nonequilibrium Complex Systems have been related to or analyzed in the context of the so-called Complex Ginzburg-Landau equation (CGL). What is the difference between the physics underlying the usual amplitude description for stationary patterns and the one underlying the CGL? Why are there many more stable coherent structures [pulses, sources (holes), sinks] possible in systems described by the CGL than in systems exhibiting a stationary bifurcation, and what is their relation, if any, to the chaotic behavior that is characteristic of the CGL in some parameter regimes? The organizers of this workshop have asked me to try to provide some answers to these questions for the non-expert, someone with an interest in pattern formation but who has not had an introduction to the CGL before or who has not followed the recent developments in this field. Since there are several very recent review papers on this subject[3],[8],[9],[12] where a more thorough and detailed discussion can be found, I will confine myself here to a brief low-level introduction, in which I try to paint some of the main ideas with broad strokes. I stress that I do not pretend to give a balanced review — this chapter is extremely sketchy and coloured by my own interests, and I urge the reader interested in learning more about this line of research to consult the papers cited above and the references therein.