Universal algebraic relaxation of velocity and phase in pulled fronts generating periodic or chaotic states

We investigate the asymptotic relaxation of so-called pulled fronts propaga ting into an unstable state, and generalize the universal algebraic velocit y relaxation of uniformly translating fronts to fronts that generate period ic or even chaotic states. A surprising feature is that such fronts also ex hibit a universal algebraic phase relaxation. For fronts that generate a pe riodic state, like those in the Swift-Hohenberg equation or in a Rayleigh-B enard experiment, this implies an algebraically slow relaxation of the patt ern wavelength just behind the front, which should be experimentally testab le.