Pattern formation in the wake of triggered pushed fronts

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.Running head: Triggered pushed fronts

Growing Stripes, with and without Wrinkles

“…Beyond the transition from perpendicular to oblique stripes, we have analyzed detachment [13,14], k y ∼ 0 [12], and k y = 0, c x ∼ 0 [11] in prior work. From this "completist" perspective, the major challenge appears to be a description of the moduli space in a vicinity of k y , c x = 0.…”

The Swift–Hohenberg Equation under Directional-Quenching: Finding Heteroclinic Connections Using Spatial and Spectral Decompositions

“…Both techniques had been seen separately in earlier works but, to the authors knowledge, have never been used simultaneously. The far/near (spatial) decomposition as done in (9) has been called by some authors far field-core decomposition: it has been a building block in the construction of multidimensional patterns in extended domains [dPKPW10], in the study of perturbation effects on the far field of multidimensional patterns [MS18,Mon18]; in combination with bifurcation techniques it is present in the context of pattern formation as one can see in the work of Scheel and collaborators (see for instance [MS17,GS16], specially [LS17] and [MS15,§5]). Similar types of decompositions have also been exploited in combination with homogenization techniques in the study and simulation of micro-structures and defects [BLBL12,BLBL15].…”

Existence of stationary fronts in a system of two coupled wave equations with spatial inhomogeneity

“…The main objective in this paper is to show the existence of solutions in the BVP In this section, using ideas from both Goh and Scheel [11] and Doelman et al [7], we show that if coupled non-zero solutions exist in the BVP (5.1-5.2) with the piecewise constant inhomogeneity ρ 0 (x; ∆) then they persist for the smooth steep inhomogeneity ρ(x; ∆, δ) when 0 < δ 1. We summarise the results in the following persistence theorem.…”

“…In order to overcome this issue, we adapt an approach by Goh and Scheel [11]. They study fronts in the complex Ginzburg-Landau equation with a smooth single step inhomogeneity and characterise this inhomogeneity with an additional Ordinary Differential Equation (ODE).…”