Modulated and alternating waves in a reaction-diffusion model with wave instability

Abstract

Various patterns of modulated waves are found beyond the onset of the wave instability in a model reaction-diffusion system with length between 1 and 3 times the basic wavelength. With periodic boundary conditions, we observe low-frequency modulation of the speed of the travelling waves. A fascinating pattern of waves that periodically change their direction of propagation along the ring is found when the ring length is between 1.05 and 1.35 times the basic wavelength. With zero-flux boundary conditions, a related pattern of modulated standing waves is found between the domains of the half-wavelength and the one-wavelength standing waves; the period of modulation contains two phases of non-stationary travelling waves, moving alternately left and right, separated by phases of short-lived asymmetric standing waves.