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The Burgers-FKPP advection-reaction-diffusion equation with cut-off

Nikola Popovic, Mariya Ptashnyk, Zak Sattar

Abstract

We investigate the effect of a Heaviside cut-off on the front propagation dynamics of the so-called Burgers-FisherKolmogoroff-Petrowskii-Piscounov (Burgers-FKPP) advection-reaction-diffusion equation. We prove the existence and uniqueness of a travelling front solution in the presence of a cut-off in the reaction kinetics and the advection term, and we derive the leading-order asymptotics for the speed of propagation of the front in dependence on the advection strength and the cut-off parameter. Our analysis relies on geometric techniques from dynamical systems theory and specifically, on geometric desingularisation, which also known as blow-up.

The Burgers-FKPP advection-reaction-diffusion equation with cut-off

Abstract

We investigate the effect of a Heaviside cut-off on the front propagation dynamics of the so-called Burgers-FisherKolmogoroff-Petrowskii-Piscounov (Burgers-FKPP) advection-reaction-diffusion equation. We prove the existence and uniqueness of a travelling front solution in the presence of a cut-off in the reaction kinetics and the advection term, and we derive the leading-order asymptotics for the speed of propagation of the front in dependence on the advection strength and the cut-off parameter. Our analysis relies on geometric techniques from dynamical systems theory and specifically, on geometric desingularisation, which also known as blow-up.

Paper Structure

This paper contains 15 sections, 18 theorems, 97 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Geometric desingularisation
  3. Dynamics in chart ("Inner region")
  4. Dynamics in chart ("Outer region")
  5. Pulled front propagation:
  6. Pushed front propagation:
  7. Singular orbit
  8. Proof of Theorem \ref{['Myfirstthm']}
  9. Persistence of
  10. Leading-order asymptotics of
  11. Pulled front propagation:
  12. Pushed front propagation:
  13. Numerical verification
  14. Discussion
  15. Proof of Theorem \ref{['Myfirstthm']} without cut-off in advection

Key Result

Theorem 1

2021ZaMP...72..163M Equation firstorder admits a heteroclinic connection between $Q^-$ and $Q^+$ for $c\geq c_{\rm crit}$, where Moreover, the corresponding front solution to Burger-FKPP1 is pulled when $k\leq 2$ and pushed when $k>2$. For $k>2$ and $c_{\rm crit}=\frac{k}{2}+\frac{2}{k}$, the heteroclinic connection for firstorder is given explicitly by $V(U)=-\frac{k}{2}U(1-U).$

Figures (5)

  • Figure 1: Singular geometry in chart $K_2.$
  • Figure 2: Singular geometry in chart $K_1$ for $k\leq2$.
  • Figure 3: Singular geometry in chart $K_1$ for $k>2$.
  • Figure 4: Error of the approximation of $c(\varepsilon)$ by $\hat{c}(\varepsilon)$ for $k=4$ and $\varepsilon\in[10^{-4},10^{-2}]$.
  • Figure 5: Numerical difference between $c(\varepsilon)$ and $\gamma(\varepsilon)$ (green) for $k=4$ and $\varepsilon\in[10^{-4},10^{-2}]$; the error $|c(\varepsilon)-\hat{c}(\varepsilon)|$ is plotted for comparison (red).

Theorems & Definitions (36)

  • Theorem 1
  • Theorem 2
  • Remark 3
  • Remark 4
  • Lemma 5
  • Lemma 6
  • Proposition 7
  • proof
  • Lemma 8
  • Lemma 9
  • ...and 26 more