Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Propagation reversal on trees in the large diffusion regime

Hermen Jan Hupkes, Mia Jukic

Abstract

In this work we study travelling wave solutions to bistable reaction diffusion equations on bi-infinite $k$-ary trees in the continuum regime where the diffusion parameter is large. Adapting the spectral convergence method developed by Bates and his coworkers, we find an asymptotic prediction for the speed of travelling front solutions. In addition, we prove that the associated profiles converge to the solutions of a suitable limiting reaction-diffusion PDE. Finally, for the standard cubic nonlinearity we provide explicit formula's to bound the thin region in parameter space where the propagation direction undergoes a reversal.

Propagation reversal on trees in the large diffusion regime

Abstract

In this work we study travelling wave solutions to bistable reaction diffusion equations on bi-infinite -ary trees in the continuum regime where the diffusion parameter is large. Adapting the spectral convergence method developed by Bates and his coworkers, we find an asymptotic prediction for the speed of travelling front solutions. In addition, we prove that the associated profiles converge to the solutions of a suitable limiting reaction-diffusion PDE. Finally, for the standard cubic nonlinearity we provide explicit formula's to bound the thin region in parameter space where the propagation direction undergoes a reversal.
Paper Structure (12 sections, 14 theorems, 95 equations, 1 figure)

This paper contains 12 sections, 14 theorems, 95 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Dynamics on $k$-ary trees
  3. Continuum regime
  4. Spectral convergence
  5. Outlook
  6. Organization
  7. Acknowledgments
  8. Main results
  9. Fixed point problem
  10. Linear theory
  11. Proof of Proposition \ref{['prp:bates:lowBoundOnE']}
  12. Nonlinear bounds

Key Result

Proposition 2.1

Mallet-Paret1999 Suppose that (Hg) holds and pick $a \in (0,1)$ together with $d > 0$ and $k > 0$. Then there exist a speed $c=c(a,d, k)$ and a non-decreasing profile $\Phi = \Phi(a,d,k):{\mathbb R}\to {\mathbb R}$ that satisfy eqn:mr:MFDE:with:h with $h = h(d,k)$, together with the boundary conditi

Figures (1)

  • Figure 1: The red curves denote the numerically computed boundaries of the pinning region where $c(a,d,k) = 0$. The cyan curves originating from $(1/2, 0)$ represent the asymptotic prediction \ref{['eq:int:crit:diff']}, which retains its accuracy even for relatively large values of the branching factor $k > 1$.

Theorems & Definitions (27)

  • Proposition 2.1
  • Theorem 2.2: see §\ref{['sec:fix']}
  • Corollary 2.3
  • proof
  • Proposition 3.1: see §\ref{['sec:lin']}
  • Proposition 3.2: see §\ref{['sec:nl']}
  • proof : Proof of Theorem \ref{['thm:mr:main']}
  • Proposition 4.1
  • Corollary 4.2
  • proof
  • ...and 17 more