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 $\epsilon F(U,U_x,x)$. 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 $F(U,U_x,x)=H^{\mathrm{\prime }}(x)U_x+H^{\mathrm{\prime }\mathrm{\prime }}(x)U$ 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 $\pm \mathrm{\infty }$, either with slowly decreasing or increasing speeds.