Modulated Amplitude Waves and Defect Formation in the One-Dimensional Complex Ginzburg-Landau Equation

Abstract

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 $\nu \approx 0$ 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 neighboring 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.

25 pages, 18 figures

References in corpus (11)

• The World of the Complex Ginzburg-Landau Equation
• Disordered Regimes of the one-dimensional complex Ginzburg-Landau equation
• The Building Blocks of Spatiotemporal Intermittency
• Characterization of the transition from defect- to phase-turbulence
• Modulated Amplitude Waves and the Transition from Phase to Defect Chaos
• Bekki-Nozaki Amplitude Holes in Hydrothermal Nonlinear Waves
• Wound-up phase turbulence in the Complex Ginzburg-Landau equation
• Winding number instability in the phase-turbulence regime of the Complex Ginzburg-Landau Equation
• Ordered and self-disordered dynamics of holes and defects in the one-dimensional complex Ginzburg-Landau equation
• Order Parameter for the Transition from Phase to Amplitude Turbulence
• Studies of Phase Turbulence in the One Dimensional Complex Ginzburg-Landau Equation

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