We study patterns that arise in the wake of an externally triggered, spatially propagating instability in the complex Ginzburg-Landau equation.We model the trigger by a spatial inhomogeneity moving with constant speed. In the comoving frame, the trivial state is unstable to the left of the trigger and stable to the right. At the trigger location, spatio-temporally periodic wave trains nucleate. Our results show existence of coherent, " heteroclinic" profiles when the speed of the trigger is slightly below the speed of a free front in the unstable medium. Our results also give expansions for the wavenumber of wave trains selected by these coherent fronts. A numerical comparison yields very good agreement with observations, even for moderate trigger speeds. Technically, our results provide a heteroclinic bifurcation study involving an equilibrium with an algebraically double pair of complex eigenvalues. We use geometric desingularization and invariant foliations to describe the unfolding. Leading-order terms are determined by a condition of oscillations in a projectivized flow, which can be found by intersecting absolute spectra with the imaginary axis.
Bibliographical noteFunding Information:
Acknowledgements A. Scheel was partially supported by the National Science Foundation through grant NSF-DMS-0806614, NSF-DMS-1311740. This material is based upon work supported by the National Science Foundation Graduate Research Fellowship under grant NSF-GFRP-00006595. Any opinion, findings, and conclusions or recommendations expressed in this material are those of the authors(s) and do not necessarily reflect the views of the National Science Foundation.
- Geometric desingularization
- Pattern formation
- Pulled invasion fronts
- Spatial dynamics