On a variety of wave phenomena in chemical reactions

Abstract

A variety of wave phenomena are analyzed and discussed for systems in which chemical reactions and transport take place. Certain families of wave solutions of reaction-transport equations arise owing to the weak stability of a reference state to a class of perturbations. We consider both wave induction by heterogeneities and autonomous waves and seek perturbation solutions which provide the dispersion relation and the wave vector dependence of the amplitude for one-parameter families of waves characterized by the wave vector. For the case of an arbitrary reaction mechanism possessing a homogeneous steady state we derive, by use of bifurcation theory and frequency renormalization, small amplitude autonomous plane waves and standing and rotating waves. We find solutions corresponding to long wavelength waves, static structures, and phenomena existing only at intermediate frequencies and wavelengths. The theory is found to have a nonuniformity in convergence in the core region of pacemaker and spiral-like solutions. Various types of behavior are illustrated with a model for which we find analytic wave solutions. We investigate next wave extensions from a homogeneous limit cycle. For heterogeneous systems we show the relation of phase diffusion waves to kinematic waves. For homogeneous systems with arbitrary nonlinear kinetics we derive by use of singular perturbation theory long wavelength extensions for plane waves, centers, and other nonplanar autonomous phenomena and give an example of threshold excitation waves. The problem of bifurcation with symmetry breaking in a homogeneous limit cycle is analyzed with a small amplitude perturbation method.