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.

References in corpus (3)

• Formation of regular structures in the process of phase separation
• Emergence of the bifurcation structure of a Langmuir-Blodgett transfer model
• Hopf bifurcation from fronts in the Cahn-Hilliard equation

Cited by in corpus (7)

• Universal wavenumber selection laws in apical growth
• Effects of time-periodic forcing in a Cahn-Hilliard model for Langmuir-Blodgett transfer
• Spectral stability of pattern-forming fronts in the complex Ginzburg-Landau equation with a quenching mechanism
• The Swift-Hohenberg Equation under directional-quenching: finding heteroclinic connections using spatial and spectral decompositions
• Existence of stationary fronts in a system of two coupled wave equations with spatial inhomogeneity
• Wavenumber selection in coupled transport equations
• Complex pattern formation driven by the interaction of stable fronts in a competition-diffusion system