Slide 1

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 Slide 2 Slide 3 Slide 4 Slide 5 Slide 6 Slide 7 Slide 8 Slide 9 Have allowed For moving frame Have allowed For moving frame Amplitude- unstable solutions Slide 10 The Front ODE Slide 11 Simulation with: The Front ODEs (actuallly for all coherent states) The Front ODEs (actuallly for all coherent states) Slide 12 Slide 13 c2=c3=2, =0.4 =0.1 =0.2 Slide 14 as in real case generates phase gradient Actually very old: Hocking and Stewartson 1972 (prevention of blow up) Slide 15 Slide 16 Slide 17 Slide 18 Slide 19 Slide 20 Slide 21 3 - S. Popp O. Stiller, E. Kuznetsov LK 1998 No collapse for suffiently large | | and not too large b, because of phase gradient effect Slide 22 c3=15, Initial conditions: small white noise b3=3 b3=1.5 b3=0b3=-2 Slide 23 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