Patterned fronts in the wake of a parameter ramp in the complex Ginzburg Landau equation
We study of the formation of pattern-forming fronts in the presence of a rigidly-propagating parameter ramp which is slowly-varying in space. In the context of the prototypical supercritical complex Ginzburg-Landau equation, we show that not only the leading order front interface, but also the selected spatial wave number is governed by the transition of the ramp between absolute and convective instability. The slow ramp then induces a further delay of the front interface and perturbation of the selected wave number, controlled by the slow passage near a complex fold of strong- and weak-stable eigenspaces. To analyze the behavior near this fold, we perform a multiple scales analysis to predict the higher-order front interface delay in terms of zeros and poles of a complex Airy quotient inner solution. We confirm these predictions with numerical continuation of heteroclinics in the associated traveling wave equation. We also numerically characterize their spectral stability, finding accumulation of eigenvalues consistent with previous results on slow absolute spectrum. We then show the leading-order absolute/convective instability heuristic accurately describes selected wave numbers in an analogous slowly-ramped Swift-Hohenberg equation.