Stability of periodic Kuramoto-Sivashinsky waves

In this note, we announce a general result resolving the long-standing question of nonlinear modulational stability, or stability with respect to localized perturbations, of periodic travelingwave solutions of the generalized Kuramoto–Sivashinski equation, establishing that spectral modulational stability, defined in the standard way, implies nonlinear modulational stability with sharp rates of decay. The approach extends readily to other secondand higher-order parabolic equations, for example, the Cahn Hilliard equation or more general thin film models.