In its canonical form, the complex Ginzburg-Landau equation (CGL) is given by
where 
is a complex function of (rescaled)
time and space; the parameters 
and
characterise linear and nonlinear dispersion
respectively.
The equation arises in the study of nonequilibrium problems, and acts as an amplitude (or envelope or modulational) equation. It provides a universal description of weakly nonlinear spatio-temporal phenomena in extended continous media whose dispersion conforms to a very general type. We also require that the system is gauge-invariant, that is, the dynamics remain invariant to the transformation:
Such a symmetry typically arises where is the
slowly-varying amplitude of a phenomenon that is periodic in at least one variable
(space or time).