Propagation Of Waves In Periodic-Heterogeneous Bistable Systems

TL;DR

This paper analyzes wave propagation in one-dimensional periodic heterogeneous bistable media using perturbation methods, deriving equations to predict wave velocity and failure, with results validated against numerical simulations.

Contribution

It introduces analytical approaches to predict wave behavior in heterogeneous media, extending understanding beyond homogeneous systems.

Findings

Analytical methods accurately predict wave velocity and failure.

Good agreement between numerical simulations and analytical predictions.

Methods applicable for intermediate and large period lengths.

Abstract

Wave propagation in one-dimensional heterogeneous bistable media is studied using the Schl\"ogl model as a representative example. Starting from the analytically known traveling wave solution for the homogeneous medium, infinitely extended, spatially periodic variations in kinetic parameters as the excitation threshold, for example, are taken into account perturbatively. Two different multiple scale perturbation methods are applied to derive a differential equation for the position of the front under perturbations. This equation allows the computation of a time independent average velocity, depending on the spatial period length and the amplitude of the heterogeneities. The projection method reveals to be applicable in the range of intermediate and large period lengths but fails when the spatial period becomes smaller than the front width. Then, a second order averaging method must be applied. These analytical results are capable to predict propagation failure, velocity overshoot, and the asymptotic value for the front velocity in the limit of large period lengths in qualitative, often quantitative agreement with the results of numerical simulations of the underlying reaction-diffusion equation. Very good agreement between numerical and analytical results has been obtained for waves propagating through a medium with periodically varied excitation threshold.