THE CUBIC COMPLEX GINZBURG-LANDAU EQUATION FOR A BACKWARD BIFURCATION

The one-dimensional complex Ginzburg-Landau equation (CGLE) with a des tabilizing cubic nonlinearity and no saturating higher-order terms has stable bounded solutions. We consider a simple pedagogical model exhi biting qualitatively the mechanism which may suppress the divergence o f the solutions. Then we investigate the functional form of the blow-u p (collapse) solutions immediately before the divergence. From this an alysis we find analytic boundaries for the existence of collapse solut ions in the parameter space of the CGLE. A comparison with numerical s imulations demonstrates that for parameters without collapse solutions the solutions of the CGLE remain bounded for all times, Finally we di scuss the implications of our results for the solutions of the CGLE wh en saturating higher-order terms are included. (C) 1998 Elsevier Scien ce B.V.