Fronts in the wake of a parameter ramp: slow passage through pitchfork and fold bifurcations

TL;DR

This paper analyzes how fronts form and evolve in the Allen-Cahn equation with slowly varying spatial heterogeneity, revealing delays caused by slow passage through bifurcations and employing advanced mathematical techniques to rigorously locate invariant manifold intersections.

Contribution

It provides a rigorous analysis of front formation in heterogeneous Allen-Cahn equations, introducing projective coordinates and geometric singular perturbation theory to handle slow passage through bifurcations.

Findings

Existence and stability of monotone fronts established for slowly varying ramps.

Identification of delays in front location due to slow passage through bifurcations.

Application of geometric singular perturbation theory and blow-up techniques to locate invariant manifold intersections.

Abstract

This work studies front formation in the Allen-Cahn equation with a parameter heterogeneity which slowly varies in space. In particular, we consider a heterogeneity which mediates the local stability of the zero state and subsequent pitchfork bifurcation to a non-trivial state. For slowly-varying ramps which are either rigidly propagating in time or stationary, we rigorously establish existence and stability of positive, monotone fronts and give leading order expansions for their interface location. For non-zero ramp speeds, and sufficiently small ramp slopes, the front location is determined by the local transition between convective and absolute instability of the base state and leads to an O(1) delay beyond the instantaneous pitchfork location before the system jumps to a nontrivial state. The slow ramp induces a further delay of the interface controlled by a slow-passage through a fold of strong- and weak-stable eigenspaces of the associated linearization. We introduce projective coordinates to de-singularize the dynamics near the trivial state and track relevant invariant manifolds all the way to the fold point. We then use geometric singular perturbation theory and blow-up techniques to locate the desired intersection of invariant manifolds. For stationary ramps, the front is governed by the slow passage through the instantaneous pitchfork bifurcation with inner expansion given by the unique Hastings-McLeod connecting solution of Painlev\'{e}'s second equation. We once again use geometric singular perturbation theory and blow-up to track invariant manifolds into a neighborhood of the non-hyperbolic point where the ramp passes through zero and to locate intersections.