Pattern formation in the wake of triggered pushed fronts. (28th June 2016)

• Record Type: Journal Article
• Authors: Goh, Ryan; Scheel, Arnd
• Is Part Of: Nonlinearity. Volume 29:Number 8(2016:Aug.)
• Page Start: 2196
• Page End: 2237
• Publication Date: 2016-06-28
• Subjects: pushed fronts -- heteroclinic bifurcation -- Ginzburg–Landau equation -- Cahn–Hilliard equation -- pattern formation; 35B36 -- 37C29 -- 35C07 -- 37G25; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/0951-7715/29/8/2196
• Languages: English
• ISSNs: 0951-7715

Abstract

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 phenomenon, including snaking, non-monotonic wave-number 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 wave-number 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.