We discuss the results of the interaction ofcounter-propagating pulses for two coupled complex cubic-quintic Ginzburg-Landau equations as they arise near the onset of a weakly inverted Hopf bifurcation. As a result of the interaction of the pulses we find in 1D for periodic boundary conditions (corresponding to an annular geometry) many different possible outcomes. These are summarized in two phase diagrams using the approach velocity, v, and the real part of the cubiccross-coupling, cr, of the counter-propagating waves asvariables while keeping all other parameters fixed. The novelphase diagram in the limit v → 0, cr → 0 turns out to beparticularly rich and includes bound pairs of 2 π holes aswell as zigzag bound pairs of pulses.
Bibliographical noteFunding Information:
O.D. wishes to thank the support of FAI (Project No. ICIV-001-06, U. de los Andes), FONDECYT (Project No.1070098) and Project Anillo en Ciencia y Tecnología ACT15. J.C. thanks FONDECYT (Project No.1070098) for financial support. P.G. acknowledges support from Project Anillo en Ciencia y Tecnología ACT15. H.R.B. thanks the Deutsche Forschungsgemeinschaft for partial support of his work through Sonderforschungsbereich 481 ‘Polymere und Hybridmaterialien in inneren und äußeren Feldern’.
- Traveling-wave convection
- Weakly inverted bifurcation
- Reaction-diffusion system
- Localized solutions
- Subcritical instabilities