Slowly Modulated Oscillations in Nonlinear Diffusion Processes

It is shown here that certain systems of nonlinear (parabolic) reaction-diffusion equations have solutions which are approximated by oscillatory functions in the form R(ξ - cτ )P(t^ * ) where P(t^ * ) represents a sinusoidal oscillation on a fast time scale t^ * and R(ξ - cτ ) represents a slowly-varying modulating amplitude on slow space (ξ ) and slow time (τ ) scales. Such solutions describe phenomena in chemical reactors, chemical and biological reactions, and in other media where a stable oscillation at each point (or site) undergoes a slow amplitude change due to diffusion.