Modulated amplitude waves and defect formation in the one-dimensional complex Ginzburg-Landau equation

The transition from phase chaos to defect chaos in the complex Ginzburg-Landau equation (CGLE) is related to saddle-node bifurcations of modulated amplitude waves (MAWs). First, the spatial period P of MAWs is shown to be limited by a maximum P-SN which depends on the CGLE coefficients; MAW-like structures with period larger than P-SN evolve to defects. Second, slowly evolving near-MAWs with average phase gradients v approximate to 0 and various periods occur naturally in phase chaotic states of the CGLE. As a measure for these periods, we study the distributions of spacings p between neighbouring peaks of the phase gradient. A systematic comparison of p and P-SN as a function of coefficients of the CGLE shows that defects are generated at locations where p becomes larger than P-SN. In other words, MAWs with period P-SN represent "critical nuclei" for the formation of defects in phase chaos and may trigger the transition to defect chaos. Since rare events where p becomes sufficiently large to lead to defect formation may only occur after a long transient, the coefficients where the transition to defect chaos seems to occur depend on system size and integration time, We conjecture that in the regime where the maximum period P-SN has diverged, phase chaos persists in the thermodynamic limit. (C) 2001 Published by Elsevier Science B.V.