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Travelling front solutions in a spatially heterogeneous reaction-diffusion system

M. Chirilus-Bruckner, L. van Vianen, F. Veerman

Abstract

We investigate a two-component reaction-diffusion system with a slow-fast structure and spatially varying coefficients $f_1$ and $f_2$ appearing in the slow equation. Under mild boundedness and regularity conditions on $f_1$ and $f_2$ the system is shown to exhibit bi-stability in the form of two stable stationary heterogeneous background states. These background states can be connected by stationary and travelling front solutions. Travelling fronts feature an interface that moves with a non-uniform speed through the motionless spatially varying background states it connects. We construct both the background states and stationary fronts using an extension of Fenichel theory to the non-compact case. Additionally, we establish the existence of travelling front solutions and derive a leading-order expression for the dynamic position of the moving interface through a time-dependent spatial dynamics approach. This expression takes the form of a delay-differential equation, and its accuracy is validated through numerical simulations. A key contribution of our work lies in the general treatment of $f_1$ and $f_2$, which are neither (necessarily) asymptotically small nor restricted to specific forms such as periodic or localized structures. Furthermore, our derivation of the front position formula circumvents the traditional reliance on spectral analysis, enabling us to describe front dynamics beyond bifurcations from stationary fronts. This approach has the potential to be extended to other settings in which spectral properties at onset preclude conventional reduction techniques.

Travelling front solutions in a spatially heterogeneous reaction-diffusion system

Abstract

We investigate a two-component reaction-diffusion system with a slow-fast structure and spatially varying coefficients and appearing in the slow equation. Under mild boundedness and regularity conditions on and the system is shown to exhibit bi-stability in the form of two stable stationary heterogeneous background states. These background states can be connected by stationary and travelling front solutions. Travelling fronts feature an interface that moves with a non-uniform speed through the motionless spatially varying background states it connects. We construct both the background states and stationary fronts using an extension of Fenichel theory to the non-compact case. Additionally, we establish the existence of travelling front solutions and derive a leading-order expression for the dynamic position of the moving interface through a time-dependent spatial dynamics approach. This expression takes the form of a delay-differential equation, and its accuracy is validated through numerical simulations. A key contribution of our work lies in the general treatment of and , which are neither (necessarily) asymptotically small nor restricted to specific forms such as periodic or localized structures. Furthermore, our derivation of the front position formula circumvents the traditional reliance on spectral analysis, enabling us to describe front dynamics beyond bifurcations from stationary fronts. This approach has the potential to be extended to other settings in which spectral properties at onset preclude conventional reduction techniques.

Paper Structure

This paper contains 35 sections, 1 theorem, 69 equations, 21 figures.

Table of Contents

  1. Introduction
  2. Background: Nagumo/Allen-Cahn setting.
  3. Background: Constant-coefficient reaction-diffusion model.
  4. Fast-reaction scaling.
  5. Main results and Plan of the paper
  6. Numerical illustration of travelling front solutions
  7. Related work and novelty of the present work
  8. Stationary background states
  9. Front solutions
  10. Stationary front solutions
  11. Travelling fronts with non-zero speed
  12. Initialised travelling fronts
  13. Fronts sets $\Gamma$
  14. Temporal evolution of $(\partial_s U, \partial_s V)$.
  15. Positive feedback $\alpha < 0$
  16. ...and 20 more sections

Key Result

Lemma 4.3

Let $a,b,c,d\in \mathbb{R}$ and let $\ell_i$, $i=1,2,3$ be $3$ lines of the form $\ell_1 : (v,q)= [(a,b)+(c,d)] +\mathbb{R}(1,1)$, $\ell_2 :(v,q) =[-(a,b)+ (c,d)] + \mathbb{R}( 1,-1)$ and $\ell_3:v=v^*$. Then $\ell_i$ intersect in a single point if and only if $c=v^* + b$.

Figures (21)

  • Figure 1: Numerical computation of solutions of \ref{['eq:PDE_model']}. Upper panels. Snapshot in time of the (black) $U$-profile along with (red) background states (left upper panel) and (black) $V$-profile along with (red) background states (right upper panel). Lower panel. Density plot for the $V$-profile along with (black) position $z(t)$, defined by the unique zero of the $U$-component, see Definition \ref{['def:position']}. Details of numerical computation. Coefficients $f_1, f_2$ from \ref{['eq:coefficients_numerics']}; Initial Conditions: $u(x,0) = \mathrm{tanh}(x/\varepsilon), v(x,0)=0.1$; Boundary conditions: homogeneous Neumann; Parameter settings: $\varepsilon = 0.15, \alpha = 0.94, \gamma = 0, \hat{\tau} = 1$. Numerical solver: MATLAB (2020): "pdepe". Code is available at chirilusbruckner2026front
  • Figure 2: A visualisation of the reduced slow dynamics in a neighbourhood of $\text{BO}^+_0$ in the hyperplane $\left\{ u=+1,\, p=0\right\}$. Left: the manifold $\mathcal{M}_0^+$ in black, the singular bounded orbit $\text{BO}^+_0$ in red, the slow unstable manifold of $\text{BO}^+_0$ in purple, the slow stable manifold of $\text{BO}^+_0$ in yellow. Right: the intersection with the hyperplane $\left\{\chi = \chi_0\right\}$, with the line $\ell^{u,+}(\chi_0)$ in purple and $\ell^{s,+}(\chi_0)$ in yellow; the compact set $\mathcal{K}$ is indicated in gray.
  • Figure 3: A visualisation of the slow dynamics in a neighbourhood of $\text{BO}^+_\varepsilon$ near the hyperplane $\left\{\,u=+1,\;p=0\,\right\}$. Left: the manifold $\mathcal{M}_\varepsilon^+$ in black, the background orbit $\text{BO}^+_\varepsilon$ in red, the slow unstable manifold of $\text{BO}^+_\varepsilon$ in purple, the slow stable manifold of $\text{BO}^+_\varepsilon$ in yellow. Right: the intersection with the hyperplane $\left\{\chi = \chi_0\right\}$, with the fibre $\mathcal{F}^{u,+}(\chi_0)$ in purple and $\mathcal{F}^{s,+}(\chi_0)$ in yellow; the compact set $\mathcal{K}$ is indicated in gray.
  • Figure 4: A visual representation of a travelling front solution as described in Definition \ref{['def:front']}.
  • Figure 5: $f_2(x)$ (in black), $v_b^-(x;0)+1$ (orange) and $q_b^-(x;0)$ (yellow). The shape of $f_2$ is chosen such that it involves multiple local minima and maxima. The corresponding $q_b^-(x;0)$ also fluctuates and we expect the front velocity to change over time as well, in an interesting way. The parameter $\gamma$ is chosen to avoid the presence of stationary fronts, i.e. intersections of the graph of $q_b^-(\cdot;0)$ with the horizontal line at height $\frac{\gamma}{\alpha}=0.4$.
  • ...and 16 more figures

Theorems & Definitions (14)

  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • Definition 3.1: Front solutions
  • Definition 3.2: Entire front solutions
  • Remark 3.4
  • Definition 3.5: Initialised front solutions
  • ...and 4 more