Pattern formation in the wake of triggered pushed fronts

TL;DR

This paper investigates how external triggers influence pattern formation in unstable media, revealing complex phenomena like snaking and hysteresis through analytical and numerical methods, with applications to Ginzburg-Landau and Cahn-Hilliard models.

Contribution

It provides a rigorous analysis of triggered pushed fronts, deriving conditions for their existence and characterizing their wavenumber selection in an abstract setting.

Findings

Existence of trigger fronts proven using heteroclinic bifurcation techniques.

Leading order expansion for the selected wavenumber in terms of trigger speed.

Identification of bifurcation curves governed by spatial eigenvalues.

Abstract

Pattern-forming fronts are often controlled by an external stimulus which progresses through a stable medium at a fixed speed, rendering it unstable in its wake. By controlling the speed of excitation, such stimuli, or "triggers," can mediate pattern forming fronts which freely invade an unstable equilibrium and control which pattern is selected. In this work, we analytically and numerically study when the trigger perturbs an oscillatory pushed free front. In such a situation, the resulting patterned front, which we call a pushed trigger front, exhibits a variety of interesting phenomenon, including snaking, non-monotonic wavenumber selection, and hysteresis. Assuming the existence of a generic oscillatory pushed free front, we use heteroclinic bifurcation techniques to prove the existence of trigger fronts in an abstract setting motivated by the spatial dynamics approach. We then derive a leading order expansion for the selected wavenumber in terms of the trigger speed. Furthermore, we show that such a bifurcation curve is governed by the difference of certain strong-stable and weakly-stable spatial eigenvalues associated with the decay of the free pushed front. We also study prototypical examples of these phenomena in the cubic-quintic complex Ginzburg Landau equation and a modified Cahn-Hilliard equation.