Noise-sustained structure, intermittency and the Ginzburg-Landau equation

The time-dependent generalized Ginzburg-Landau equation is a partial differential equation that is related to many physical systems. In the stationary (i.e., laboratory) frame of reference the equation is: Partial psi/partial t = a psi - NU) (partial psi/partial x) + b (partial squared x/partial x) 2 - c (psi absolute) 2 psi where the dependent variable psi is in general complex; a, b, and c are constants which are in general complex; and NU is the group velocity. A small initial localized perturbation is considered for the equilibrium state psi = 0. A linear stability analysis reveals that there are three types of behavior which this perturbation can undergo: (1) the perturbation will be damped in any frame of reference; this behavior corresponds to the system being absolutely stable. (2) The perturbation will grow and spread such that the edges of the perturbation move in opposite directions; this behavior corresponds to the system being absolutely unstable; (3) the perturbation will be damped at any given stationary point, but a frame of reference may be found in which the perturbation is growing.