Universality of Crystallographic Pinning

TL;DR

This paper investigates the phenomenon of crystallographic pinning in reaction-diffusion equations on a discrete grid, demonstrating its occurrence in specific directions under certain nonlinear conditions using dynamical systems analysis.

Contribution

It extends understanding of crystallographic pinning by proving its presence in horizontal and vertical directions for bistable nonlinearities under a generic condition.

Findings

Crystallographic pinning occurs in horizontal and vertical directions.

Pinning is shown for bistable nonlinearities satisfying a generic condition.

The proof uses analysis of heteroclinic chains in dynamical systems.

Abstract

We study traveling waves for reaction diffusion equations on the spatially discrete domain $\Z^2$. The phenomenon of crystallographic pinning occurs when traveling waves become pinned in certain directions despite moving with non-zero wave speed in nearby directions. Mallet-Paret has shown that crystallographic pinning occurs for all rational directions, so long as the nonlinearity is close to the sawtooth. In this paper we show that crystallographic pinning holds in the horizontal and vertical directions for bistable nonlinearities which satisfy a specific computable generic condition. The proof is based on dynamical systems. In particular, it relies on an examination of the heteroclinic chains which occur as singular limits of wave profiles on the boundary of the pinning region.