Two reasons for the appearance of pushed wavefronts in the Belousov-Zhabotinsky system with spatiotemporal interaction

TL;DR

This paper proves the existence of a minimal wavefront speed in the Belousov-Zhabotinsky system with spatiotemporal interactions, identifying conditions that lead to pushed wavefronts, which are faster than the minimal speed predicted by linearization.

Contribution

It establishes the conditions under which pushed wavefronts occur in the Belousov-Zhabotinsky system with convolution-based interactions, highlighting the roles of parameter size and kernel asymmetry.

Findings

Minimal wavefront speed is within [2√(1-r), 2].

Large positive parameter b leads to pushed wavefronts.

Asymmetric kernels induce pushed wavefronts.

Abstract

We prove the existence of the minimal speed of propagation $c_*(r,b,K) \in [2\sqrt{1-r},2]$ for wavefronts in the Belousov-Zhabotinsky system with a spatiotemporal interaction defined by the convolution with (possibly, "fat-tailed") kernel $K$. The model is assumed to be monostable non-degenerate, i.e. $r\in (0,1)$. The slowest wavefront is termed pushed or non-linearly determined if its velocity $c_*(r,b,K) > 2\sqrt{1-r}$. We show that $c_*(r,b,K)$ is close to 2 if i) positive system's parameter $b$ is sufficiently large or ii) if $K$ is spatially asymmetric to one side (e.g. to the left: in such a case, the influence of the right side concentration of the bromide ion on the dynamics is more significant than the influence of the left side). Consequently, this reveals two reasons for the appearance of pushed wavefronts in the Belousov-Zhabotinsky reaction.