Propagation failure of traveling waves in a discrete bistable medium

TL;DR

This paper investigates the conditions under which traveling waves fail to propagate in a discrete bistable medium, providing exact calculations of critical points and wavefront profiles, and revealing a unique universality class distinct from smooth models.

Contribution

It offers exact analysis of propagation failure in a discrete bistable reaction-diffusion system, highlighting differences from smooth models and characterizing wave speed scaling near criticality.

Findings

Wave speed vanishes logarithmically near the transition

Exact critical points and wavefront profiles are derived

The model belongs to a different universality class than Nagumo

Abstract

Propagation failure (pinning) of traveling waves is studied in a discrete scalar reaction-diffusion equation with a piecewise linear, bistable reaction function. The critical points of the pinning transition, and the wavefront profile at the onset of propagation are calculated exactly. The scaling of the wave speed near the transition, and the leading corrections to the front shape are also determined. We find that the speed vanishes logarithmically close to the critical point, thus the model belongs to a different universality class than the standard Nagumo model, defined with a smooth, polynomial reaction function.