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Multi-front dynamics in spatially inhomogeneous Allen-Cahn equations

Robbin Bastiaansen, Arjen Doelman, Tasso J. Kaper

TL;DR

This work develops a comprehensive framework to understand how spatial heterogeneity shapes front dynamics in the one-dimensional Allen–Cahn equation with small forcing $U_t = U_{xx} + U - U^3 + \varepsilon F(U,U_x,x)$. By combining Melnikov theory, invariant-manifold persistence, and matched asymptotics, it derives existence and stability criteria for stationary one- and two-front patterns, and reduces multi-front dynamics to an explicit $N$-dimensional ODE system governing front positions. It then analyzes two principal forcing classes—localized topographies and spatially periodic forcing—showing pinning and coarsening suppression in periodic settings, and the formation of slowly traveling front trains under localization; it extends the analysis to $N$-front patterns, obtaining existence counts and stability properties, with numerical simulations illustrating the rich phenomenology. The results provide a principled view of how heterogeneity can stabilize, pin, or dynamically reorganize patterns, with potential applications to ecological and chemical pattern formation where spatial structure is intrinsic. Overall, the paper offers a rigorous, scalable approach to predicting and analyzing complex multi-front dynamics in inhomogeneous reaction-diffusion systems, supported by explicit formulas and illustrative simulations.

Abstract

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.

Multi-front dynamics in spatially inhomogeneous Allen-Cahn equations

TL;DR

This work develops a comprehensive framework to understand how spatial heterogeneity shapes front dynamics in the one-dimensional Allen–Cahn equation with small forcing . By combining Melnikov theory, invariant-manifold persistence, and matched asymptotics, it derives existence and stability criteria for stationary one- and two-front patterns, and reduces multi-front dynamics to an explicit -dimensional ODE system governing front positions. It then analyzes two principal forcing classes—localized topographies and spatially periodic forcing—showing pinning and coarsening suppression in periodic settings, and the formation of slowly traveling front trains under localization; it extends the analysis to -front patterns, obtaining existence counts and stability properties, with numerical simulations illustrating the rich phenomenology. The results provide a principled view of how heterogeneity can stabilize, pin, or dynamically reorganize patterns, with potential applications to ecological and chemical pattern formation where spatial structure is intrinsic. Overall, the paper offers a rigorous, scalable approach to predicting and analyzing complex multi-front dynamics in inhomogeneous reaction-diffusion systems, supported by explicit formulas and illustrative simulations.

Abstract

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 . In the first part, we perform an analysis of the existence and stability of stationary one-, two- and -front patterns for general spatial heterogeneity . In addition, we explicitly determine the -th order system of ODEs that governs the evolution of the front positions of general -front patterns to leading order. In the second part, we focus on a particular class of spatial heterogeneities where with 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 -front patterns, and show that these are unstable for . We find instead slowly evolving `trains' of -fronts that collectively travel to , either with slowly decreasing or increasing speeds.

Paper Structure

This paper contains 28 sections, 8 theorems, 190 equations, 23 figures.

Table of Contents

  1. Introduction
  2. Zero-front solutions: the persistence of the background states
  3. Stationary one-front solutions in the inhomogeneous Allen-Cahn equation
  4. One-front solutions in the homogeneous Allen-Cahn equation
  5. The existence of stationary one-fronts
  6. Linear stability of stationary one-front solutions
  7. Stationary one-front solutions in topographically driven and in spatially periodic systems
  8. One-front solutions arising from weak topographical effects
  9. One-front solutions in systems with spatially-periodic heterogeneities
  10. Stationary two-front solutions in the inhomogeneous Allen-Cahn equation
  11. The existence of stationary two-front solutions
  12. Linear stability of stationary two-front solutions
  13. Stationary two-fronts in topographically driven systems and spatially periodic systems
  14. Two-front solutions solutions arising from weak topographical effects
  15. Two-front solutions in systems with spatially-periodic heterogeneities
  16. ...and 13 more sections

Key Result

Theorem 2.1

Consider the weakly heterogeneous Allen-Cahn equation eq:mainEquation with bounded inhomogeneity $F(U,U_x,x)$. Let the Melnikov function $\mathcal{R}(\phi)$ be given by eq:FredholmCondition1Front with $u_h = u_{\rm up}$ and assume that $\mathcal{R}(\phi)$ has a non-degenerate zero at $\phi = \phi_\a

Figures (23)

  • Figure 1: Simulations of multi-front dynamics in \ref{['eq:mainEquation']} (under homogeneous Neumann boundary conditions) with topographical inhomogeneities $H(x)$ given by \ref{['eq:Ftopography']} (with $\varepsilon = 0.1$); (a), (b), (c): starting from the two-front initial condition $U(x,0) = \tanh(x + 4) - \tanh(x - 4) - 1$ on the $x$-interval $(-10,10)$; (d), (e), (f): with three-front initial condition $U(x,0) = \tanh(x + 7) - \tanh(x - 1) + \tanh(x - 9.1)$ on $(-25,25)$, plotted on $(-15,15)$. The color indicates the magnitude of $u$.
  • Figure 2: Examples of numerically attracting (and thus numerically stable) stationary multi-front patterns in the forced Allen-Cahn equation \ref{['eq:mainEquation']} with spatially periodic inhomogeneous term $F(U,V,x)$ given by \ref{['eq:canonicalExample']} with $f_1(x) = \cos(\frac{4}{15}\pi x)$, $f_2(x) \equiv f_3(x) \equiv 0$, and $\varepsilon = 0.1$. (Simulations done for $x \in (-300,300)$ with homogeneous Neumann boundary conditions; patterns are (numerically) stable based on simulation up to $T = 10^{12}$.)
  • Figure 3: Simulations of \ref{['eq:mainEquation']} with five-front initial condition $U(x,0) = \tanh(x - 1) - \tanh(x - 15) + \tanh(x - 25) - \tanh(x - 37) + \tanh(x - 48)$ and $\varepsilon = 0.1$ on the interval $(-500,1500)$ (plotted on relevant subintervals) with homogeneous Neumann boundary conditions (a) A spatially periodic inhomogeneity \ref{['eq:canonicalExample']} with $f_1(x) = \sin(\frac{1}{5} x)$, $f_2(x) \equiv f_3(x) \equiv 0$. (b) An algebraically decaying spatially localized topographical inhomogeneity $H(x) = -H_{\rm alg}(x; p)$ with $0 < p = 0.25$\ref{['eq:Ftopography']}, \ref{['eq:defHuni-alg']}. (c) The inhomogeneity $H(x) = -H_{\rm alg}(x; p)$ with $p = -0.50 \in (-1,0)$.
  • Figure 4: The zero-front background states $u_+^\varepsilon(x)$ for inhomogeneous equation \ref{['eq:mainEquation']} with topographical inhomogeneities \ref{['eq:Ftopography']}. (a) $H(x) = H_{\rm alg}(x;4.0)$, (b) $H(x) = H_{\rm alg}(x;0.0)$, (c) $H(x) = \sin 2x$ and $\varepsilon = 0.25$. Plots are obtained via numerical simulations; see also remark \ref{['remark:numerics']}
  • Figure 5: (a) The phase portrait for the unperturbed ($\varepsilon =0$) planar system \ref{['eq:stationaryODEsystem']} (determined by the level sets of $\mathcal{H}(u,u_x)$\ref{['eq:defHamH']}), (b) the unperturbed one-front pattern $u_\textrm{up}(x;0)$\ref{['eq:heteroclinicSolution']} as stable solution of \ref{['eq:mainEquation']} with $\varepsilon = 0$ and (c-d) (stable) stationary patterns in \ref{['eq:mainEquation']} with $\varepsilon = 0.5$ with (c) a 'localized' and (d) a spatially periodic topography \ref{['eq:Ftopography']} ((c) $H(x) = H_{\rm alg}(x;-0.6)$, (d) $H(x) = \sin 2x$). (Note that we have chosen $p < 0$ in (c) to better illustrate the impacts of the heterogeneity, whereas with $p < 0$ the topography is not really localised (see also Sec. \ref{['sec:Discussion']}).
  • ...and 18 more figures

Theorems & Definitions (19)

  • Remark 1.1
  • Remark 1.2
  • Remark 1.3: numerical simulations
  • Theorem 2.1
  • Remark 3.1
  • Remark 3.2
  • Theorem 3.3
  • Remark 3.4
  • Remark 3.5
  • Remark 4.1
  • ...and 9 more