Triggered Fronts in the Complex Ginzburg Landau Equation

Abstract

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 wavetrains 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 wavetrains 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

Pattern‐forming fronts in a Swift–Hohenberg equation with directional quenching — parallel and oblique stripes

“…It turns out that for isotropic systems, such transversely modulated invasion speeds are always slower than non‐modulated invasion, for all ; [, § 7]. We therefore expect that, proving existence of transversely modulated invasion fronts together with a gluing result as in would give existence of oblique fronts up to , for a rather general class of pattern‐forming equations. The fact that oblique fronts exist up to a maximal speed slower than the maximal speed of parallel fronts then explains the fact that, across many experiments from reaction–diffusion settings to phase separation processes, one observes stripes parallel to the quenching line at large speeds.…”

Self-organized dip-coating patterns of simple, partially wetting, nonvolatile liquids

“…This externally imposed front velocity is a control parameter of the system. Examples for investigations of such triggered pattern formation range from the experimentally and theoretically investigated structure formation in Langmuir-Blodgett films [29][30][31][32][33][34], over the study of Cahn-Hilliard-type model equations in one (1D) and two (2D) dimensions for externally quenched phase separation (e.g., by a moving temperature jump for films of polymer blends or binary mixtures) [35][36][37] to the rigorous mathematical analysis of trigger fronts in a complex Ginzburg-Landau equation as well as in a Cahn-Hilliard and an Allen-Cahn equation [38][39][40]. In the aforementioned systems, a switch from a linearly stable to an unstable state takes place at a certain position within the considered domain.…”

Growing Stripes, with and without Wrinkles

“…Near the detachment, the results in [13] establish corrections to the wavenumber based on absolute spectra. Based on these predictions, one concludes in this regime near the linear spreading speed, that the transition from oblique to perpendicular stripes occurs at leading order when the absolute spectrum, computed in the co-moving frame, possesses a triple point at λ = 0.…”

“…Our main analytical contribution is the analysis of a heteroclinic bifurcation at c x = 0 which allows us to establish existence of solutions growing slanted stripes, and predict leading-order asymptotics for the angle. We also make extensively use of predictions for wavenumbers based on pinched double roots and absolute spectra [16,32,31,13].…”