Traveling Waves in Bistable Reaction-Diffusion Cellular Automata

TL;DR

This paper explores traveling wave phenomena in bistable reaction-diffusion cellular automata, revealing unique wave behaviors, speed restrictions, and the existence of higher-order waves with periodic profile changes.

Contribution

It introduces the concept of traveling waves in discrete cellular automata systems and characterizes their properties, including speed limitations and the existence of higher-order waves.

Findings

High diffusion parameters restrict wave speeds to slow regimes.

Pinned waves exist for weak diffusion but do not cover all wave types.

Higher-order waves exhibit periodic profile changes and are unique to cellular automata.

Abstract

We describe various types of traveling fronts of bistable reaction-diffusion cellular automata. These dynamical systems with discrete time, space, and state spaces can be seen as fully discrete versions of widely studied bistable reaction-diffusion equations. We show that moving traveling waves for high diffusion parameters are restricted to slow speeds and their profiles are interestingly not unique. Pinned waves always exist for weak diffusion as in the case of lattice equations but do not complement parametric region of moving traveling waves. The remaining parameter domain is dominated by waves which are unique to cellular automaton settings. These higher-order traveling waves move and periodically change profile at the same time.