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

Abstract

The work proposes and studies a model for one-dimensional spatially extended systems, which involve 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. Moreover, the Swift-Hohenberg equation is derived from a general energy functional. 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 unusual 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 nonhomogeneous material.