On the Evolution of Periodic Plane Waves in Reaction-Diffusion Systems of $\lambda $–$\omega$ Type

Abstract

$\lambda $–$\omega$ systems are a class of simple reaction-diffusion equations with a limit cycle in the reaction kinetics. The author considers the solution of the system given by $\lambda ( r ) = \lambda _0 - r^p $, $\omega ( r ) = \omega_0 - r^p $ on a semi-infinite spatial domain with initial data decaying exponentially across the domain. Numerical evidence is presented, showing that this initial condition induces a wave front moving across the domain, with periodic plane waves behind the front. These periodic waves can move in either direction, depending on the parameter values. The author uses intuitive criteria to derive an expression for the speed of the advancing front, and by reducing the system to ordinary differential equations for similarity solutions, the amplitude and speed of the periodic plane waves are determined. The speed of these periodic waves varies continuously with the initial data. Perturbation theory is then used to obtain an analytical approximation to the solutions. These anal...