Hyperbolic Reaction−Diffusion Equations, Patterns, and Phase Speeds for the Brusselator

Abstract

In this paper, we study the hyperbolic reaction−diffusion equations for the irreversible Brusselator. We show that the phase speed of traveling oscillating chemical waves can be obtained from the linearized hyperbolic reaction−diffusion equations. The Luther-type speed formula is obtained in the lowest-order approximation. We also solve the hyperbolic reaction−diffusion equations for two spatial dimensions and study the evolution of patterns formed. The patterns can evolve from a symmetric to a chaotic form as the reaction−diffusion number is varied. The two-dimensional power spectra of such chaotic-looking spatial patterns are shown still to preserve some symmetry in the two-dimensional reciprocal space although the cross sections of the power spectra have the signatures of chaotic patterns. It is interesting that even chaotic patterns have some sort of symmetry in the two-dimensional reciprocal space−Fourier space.