  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 traveling-wave solutions of the
generalized Kuramoto-Sivashinsky 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 second- and higher-order parabolic equations, for example, the
Cahn-Hilliard equation or more general thin film models.
