Stability of Periodic Traveling Waves for Nonlinear Dispersive Equations

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. We then discuss when the associated linearized equation admits solutions exponentially growing in time. The proof uses variational properties of the equation.