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 perturbation. Here, we determine time-asymptotic behavior under such perturbations, showing that solutions consist to leading order of a modulation whose parameter evolution is governed by an associated Whitham averaged equation. 
Mathew Johnson, Pascal Noble, L. Miguel Rodrigues, Kevin Zumbrun
math.AP