Modulated amplitude waves and defect formation in the one-dimensional complex Ginzburg-Landau equation. (English) Zbl 0996.35071

Summary: 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 \(P_{\text{SN}}\) which depends on the CGLE coefficients; MAW-like structures with period larger than \(P_{\text{SN}}\) evolve to defects. Second, slowly evolving near-MAWs with average phase gradients \(\nu\approx 0\) 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 \(P_{\text{SN}}\) 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 \(P_{\text{SN}}\) 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 \(P_{\text{SN}}\) has diverged, phase chaos persists in the thermodynamic limit.


35Q55 NLS equations (nonlinear Schrödinger equations)
76F20 Dynamical systems approach to turbulence
35B32 Bifurcations in context of PDEs


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