Perfect stationary solutions of reaction-diffusion equations on lattices and regular graphs

TL;DR

This paper introduces perfect stationary solutions for reaction-diffusion equations on lattices and graphs, showing their properties, existence, and applications, including uncountably many irregular solutions and a bistable reaction-diffusion application.

Contribution

It defines perfect stationary solutions, connects them with perfect colorings, and proves their existence on various grids, extending the understanding of stationary solutions in reaction-diffusion systems.

Findings

Existence of perfect stationary solutions on square, triangular, and hexagonal grids.

Uncountably many two-valued, highly irregular stationary solutions.

Application to bistable reaction-diffusion equations on square grids.

Abstract

Reaction-diffusion equations on infinite graphs can have an infinite number of stationary solutions. These solutions are generally described as roots of a countable system of algebraic equations. As a generalization of periodic stationary solutions we present perfect stationary solutions, a special class of solutions with finite range in which the neighborhood values are determined precisely by the value of the central vertex. The focus on the solutions which attain a finite number of values enables us to rewrite the countable algebraic system to a finite one. In this work, we define the notion of perfect stationary solutions and show its elementary properties. We further present results from the theory of perfect colorings in order to prove the existence of the solutions in the square, triangular and hexagonal grids; as a byproduct, the existence of uncountable number of two-valued stationary solutions on these grids is shown. These two-valued solutions can form highly aperiodic and highly irregular patterns. Finally, an application to a bistable reaction-diffusion equation on a square grid is presented.