Stability of Periodic Traveling Waves for Nonlinear Dispersive Equations

Abstract

We study the stability and instability of periodic traveling waves for Korteweg--de Vries-type equations with fractional dispersion and related, nonlinear dispersive equations. We show that a local constrained minimizer for a suitable variational problem is nonlinearly stable to period preserving perturbations, provided that the associated linearized operator enjoys a Jordan block structure. We then discuss when the linearized equation admits solutions exponentially growing in time.