Generalization of the reaction-diffusion, the Swift-Hohenberg and the Kuramoto-Sivashinsky equation and effects of finite propagation speeds

TL;DR

This paper introduces a unified one-dimensional model incorporating nonlocal interactions and finite propagation speeds, extending classical equations and revealing new stability phenomena relevant to non-Fourier heat conduction.

Contribution

It generalizes reaction-diffusion, Swift-Hohenberg, and Kuramoto-Sivashinsky equations to include finite speeds, uncovering new instabilities and explaining recent experimental observations.

Findings

Identification of critical propagation speeds for stability

Discovery of novel instability types in extended equations

Application to non-Fourier heat conduction phenomena

Abstract

The work proposes and studies a one-dimensional model, which involves nonlocal interactions and finite propagation speed. It shows that the general reaction-diffusion equation, the Swift-Hohenberg equation and the general Kuramoto-Sivashinsky equation represent special cases of the proposed model for limited spatial interaction ranges and for infinite propagation speeds. After a detailed validity study on the generalization conditions, the three equations are extended to involve finite propagation speeds. Moreover linear stability studies of the extended equations reveal critical propagation speeds and novel types of instabilities in all three equations. In addition, an extended diffusion equation is derived and studied in some detail with respect to finite propagation speeds. The extended model allows for the explanation of recent experimental results on non-Fourier heat conduction in non-homogeneous material.