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Fronts in the cubic and quintic complex Ginzburg-Landau equation - Linear fronts in the supercritical case (cubic CGLe) - Normal and retracting fronts in subcritical bifurcations (quintic CGL2) - Spatiotemporal intermittency - Localized states: pulses and holes - Retracting fronts in supercritical Hopf bifurcations - Collapse (finite-time blow up) - Conclusions Work on retracting fronts together with P. Coullet (Chaos, submitted) Benasque, September 2003

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Have allowed For moving frame Have allowed For moving frame Amplitude- unstable solutions

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The “Front ODE”

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Simulation with: The “Front ODEs” (actuallly for all coherent states) The “Front ODEs” (actuallly for all coherent states)

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c2=c3=2, ß=0.4 µ=0.1 µ=0.2

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as in real case generates phase gradient Actually very old: Hocking and Stewartson 1972 (prevention of blow up)

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3 - S. Popp O. Stiller, E. Kuznetsov LK 1998 No collapse for suffiently large | | and not too large b, because of phase gradient effect

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c3=15, Initial conditions: small white noise b3=3 b3=1.5 b3=0b3=-2

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Retracting Fronts and their consequences are a very general and robust phenomenon (other models, far away from threshold!). Only need nonlinear dispersion. Have various important consequences: - basic state absolutely stable for negative - spatiotemporal intemittency for positive (no hysteresis in the subcritical case) - localized structures (pulses and holes) - prevention of blow-up: for subcritical bifurcations the relevant solutions may bifurcate supercritically Concluding Remarks

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