Weak and strong interaction of excitation kinks in scalar parabolic equations

TL;DR

This paper investigates the interactions of excitation kinks in scalar reaction-diffusion equations, revealing geometric properties, collision dynamics, and metastable behavior, and contrasting these with cellular automata models.

Contribution

It provides a detailed analysis of kink interactions in PDEs, including geometric bounds, collision data, and metastable dynamics, with a rigorous reduction to ODEs.

Findings

Periodic kink sequences become asymptotically equidistant.

Distances between kinks diverge over time due to diffusion.

Diffusion causes a loss of positional information, unlike cellular automata.

Abstract

Motivated by studies of the Greenberg-Hastings cellular automata (GHCA) as a caricature of excitable systems, in this paper we study kink-antikink dynamics in the perhaps simplest PDE model of excitable media given by the scalar reaction diffusion-type $\theta$-equations for excitable angular phase dynamics. On the one hand, we qualitatively study geometric kink positions using the comparison principle and the theory of terraces. This yields the minimal initial distance as a global lower bound, a well-defined sequence of collision data for kinks- and antikinks, and implies that periodic pure kink sequences are asymptotically equidistant. On the other hand, we study metastable dynamics of finitely many kinks using weak interaction theory for certain analytic kink positions, which admits a rigorous reduction to ODE. By blow-up type singular rescaling we show that distances become ordered in finite time, and eventually diverge. We conclude that diffusion implies a loss of information on kink distances so that the entropic complexity based on positions and collisions in the GHCA does not simply carry over to the PDE model.