When is the weakly nonlinear evolution of a localized disturbance governedby the Ginzburg-Landau equation?

The evolution of weakly nonlinear modulated disturbances in marginally unst able systems has been the subject of intensive study. In a wide range of pr oblems it has been proposed that such disturbances are described by the (co mplex) Ginzburg-Landau equation, or in the case of two-dimensional modulati ons in certain systems, by the Davey-Stewartson equations with complex coef ficients. In this paper, we reexamine the evolution of an initially linear localized disturbance into the weakly nonlinear regime. We show that for on e-dimensional modulations, the weakly nonlinear evolution is governed, init ially at least, by the Landau equation, rather than the Ginzburg-Landau equ ation. For two-dimensional modulations, the evolution is governed by a redu ced form of the Davey-Stewartson equations. We apply our revised theory to Poiseuille-Couette flow. For both one- and t wo-dimensional modulations ave find that a localized disturbance can always grow to order-one amplitude if its initial magnitude is sufficiently small (contrary to earlier predictions).