Rattling in spatially discrete diffusion equations with hysteresis

TL;DR

This paper investigates a reaction-diffusion model with hysteresis on a lattice, revealing a propagating microstructure called 'rattling' and analyzing its speed based on model parameters and initial data.

Contribution

It introduces the concept of rattling microstructure in discretized hysteretic reaction-diffusion equations and analyzes its propagation characteristics.

Findings

Identification of rattling microstructure in lattice models

Derivation of propagation speed depending on hysteresis parameters

Insight into effects of nontransverse initial data

Abstract

The paper treats a reaction-diffusion equation with hysteretic nonlinearity on a one-dimensional lattice. It arises as a result of the spatial discretization of the corresponding continuous model with so-called nontransverse initial data and exhibits a propagating microstructure --- which we call {\em rattling} --- in the hysteretic component of the solution. We analyze this microstructure and determine the speed of its propagation depending on the parameters of hysteresis and the nontransversality coefficient in the initial data.