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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.

35Q55NLS-like (nonlinear Schrödinger) equations
76F20Dynamical systems approach to turbulence
35B32Bifurcation (PDE)
Full Text: DOI arXiv
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