Traveling waves for a model of the Belousov-Zhabotinsky reaction

TL;DR

This paper analyzes traveling wave solutions in a mathematical model of the Belousov-Zhabotinsky reaction, exploring stability, uniqueness, and speed estimates for wavefronts, including effects of delay and bistability.

Contribution

It introduces the concept of regular super-solutions to improve bounds on wave propagation speeds and characterizes wavefronts in monostable and bistable regimes, including delayed systems.

Findings

Uniqueness of monotone wavefronts for certain parameters

Improved upper bounds for minimal wave speed

Analysis of wavefronts in delayed and bistable regimes

Abstract

Following J.D. Murray, we consider a system of two differential equations that models traveling fronts in the Noyes-Field theory of the Belousov-Zhabotinsky (BZ) chemical reaction. We are also interested in the situation when the system incorporates a delay $h\geq 0$. As we show, the BZ system has a dual character: it is monostable when its key parameter $r \in (0,1]$ and it is bistable when $r >1$. For $h=0, r\not=1$, and for each admissible wave speed, we prove the uniqueness of monotone wavefronts. Next, a concept of regular super-solutions is introduced as a main tool for generating new comparison solutions for the BZ system. This allows to improve all previously known upper estimations for the minimal speed of propagation in the BZ system, independently whether it is monostable, bistable, delayed or not. Special attention is given to the critical case $r=1$ which to some extent resembles to the Zeldovich equation.