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Pushed fronts in a Fisher-KPP-Burgers system using geometric desingularization

Matt Holzer, Matthew Kearney, Samuel Molseed, Katie Tuttle, David Wigginton

Abstract

We study traveling fronts in a system of one dimensional reaction-diffusion-advection equations motivated by problems in reactive flows. In the limit as a parameter tends to infinity, we construct the approximate front profile and determine the leading order expansion for the selected wavespeed. Such fronts are often constructed as transverse intersections of stable and unstable manifolds of the traveling wave differential equation. However, a re-scaling of the dependent variable leads to a lack of hyperbolicity for one of the end states making the definition of one such manifold unclear. We use geometric blow-up techniques to recover hyperbolicity and following an analysis of the blown up vector field are able to show the existence of a traveling front with a leading order expansion of its speed.

Pushed fronts in a Fisher-KPP-Burgers system using geometric desingularization

Abstract

We study traveling fronts in a system of one dimensional reaction-diffusion-advection equations motivated by problems in reactive flows. In the limit as a parameter tends to infinity, we construct the approximate front profile and determine the leading order expansion for the selected wavespeed. Such fronts are often constructed as transverse intersections of stable and unstable manifolds of the traveling wave differential equation. However, a re-scaling of the dependent variable leads to a lack of hyperbolicity for one of the end states making the definition of one such manifold unclear. We use geometric blow-up techniques to recover hyperbolicity and following an analysis of the blown up vector field are able to show the existence of a traveling front with a leading order expansion of its speed.
Paper Structure (18 sections, 5 theorems, 60 equations, 3 figures)

This paper contains 18 sections, 5 theorems, 60 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Traveling wave equation
  3. Front and wavespeed selection
  4. Pushed and pulled fronts
  5. Constructing pushed fronts: singular perturbations
  6. Previous work
  7. Proof strategy
  8. Preliminaries
  9. Reduction to the singular limit $\varepsilon=0$
  10. A quasi-homogeneous blow-up of the nilpotent fixed point $(1,c)$
  11. Chart $K_\varepsilon$
  12. Chart $K_W$
  13. The invariant subspace $r_2=0$
  14. The invariant subspace $\varepsilon_2=0$
  15. The transition map
  16. ...and 3 more sections

Key Result

Theorem 1.1

bramburger21 System (eq:RDA) with $\nu=0$ has traveling front solutions connecting the stable state $(1,\tilde{c}+\rho-\sqrt{\tilde{c}^2+\rho^2})$ to $(0,0)$ for any $\tilde{c}\geq \tilde{c}^*(\rho)$. These fronts are monotonic and for $\rho\to\infty$ the critical speed $\tilde{c}^*(\rho)$ satisfies

Figures (3)

  • Figure 1: The heteroclinic orbit obtained for the reduced ($\varepsilon=0$) system connecting the fixed point $(1,c)$ to the origin with $c=c^*(0)=\sqrt[3]{\frac{3}{2}}$.
  • Figure 2: The dynamics in chart $K_\varepsilon$ for $c= \sqrt[3]{\frac{3}{2}}$. The system is Hamiltonian and there exists an explicit expression for the graph of the unstable manifold that exists for $T_1<0$. The unstable manifold in upper left quadrant will end up defining the singular orbit in the blow-up system.
  • Figure 3: The relevant dynamics of the blown up fixed point at $(1,c)$. The $\varepsilon=0$ subspace is invariant and we have shown the existence of a heteroclinic connecting this fixed point to the origin in Section \ref{['sec:hetero']}. This is depicted by the red curve. On the upper half of the sphere two fixed points are shown. The one at the north pole is an artificial fixed point that arises when the singularity occurring at $W=c$ in (\ref{['eq:scale1']}) is removed. The other fixed point corresponds to the steady state $(1,w_-(0))$. In the blown-up coordinates this fixed point is hyperbolic and we are able to track its unstable manifold and show that it is heteroclinic to the fixed point at the equator of the sphere. Obtaining estimates on the dynamics for $0<\varepsilon\ll 1$ we are able to show that the unstable manifold of $(1,w_-(\varepsilon))$ is $\mathcal{O}(\varepsilon)$ close to the reduced heteroclinic shown in red.

Theorems & Definitions (10)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1
  • Remark 2.2
  • Lemma 3.1
  • proof
  • Lemma 4.1
  • Remark 4.2
  • proof
  • Remark 4.3