Nonlocalized modulation of periodic reaction diffusion waves: The Whitham equation

In a companion paper, we established nonlinear stability with detailed diffusive rates of decay of spectrally stable periodic traveling-wave solutions of reaction diffusion systems under small perturbations consisting of a nonlocalized modulation plus a localized (L ¹) perturbation. Here, we determine time-asymptotic behavior under such perturbations, showing that solutions consist of a leading order of a modulation whose parameter evolution is governed by an associated Whitham averaged equation.