Multi-front dynamics in spatially inhomogeneous Allen-Cahn equations

Recent studies of biological, chemical, and physical pattern-forming systems have started to go beyond the classic `near onset' and `far from equilibrium' theories for homogeneous systems to include the effects of spatial heterogeneities. In this article, we build a conceptual understanding of the impact of spatial heterogeneities on the pattern dynamics of reaction-diffusion models. We consider the simplest setting of an explicit, scalar, bi-stable Allen-Cahn equation driven by a general small-amplitude spatially-heterogeneous term $\varepsilon F(U,U_x,x)$. In the first part, we perform an analysis of the existence and stability of stationary one-, two- and $N$-front patterns for general spatial heterogeneity $F(U,U_x,x)$. In addition, we explicitly determine the $N$-th order system of ODEs that governs the evolution of the front positions of general $N$-front patterns to leading order. In the second part, we focus on a particular class of spatial heterogeneities where $F(U,U_x,x) = H'(x) U_x + H''(x) U$ with $H$ either spatially periodic or localised. For spatially periodic heterogeneities, we show that the fronts of a multi-front pattern will get `pinned' if the distances between successive fronts are sufficiently large, {\it i.e.}, the multi-front pattern is attracted to a nearby stable stationary multi-front pattern. For localised heterogeneities, we determine all stationary $N$-front patterns, and show that these are unstable for $N > 1$. We find instead slowly evolving `trains' of $N$-fronts that collectively travel to $\pm \infty$, either with slowly decreasing or increasing speeds.