  We consider the stability and instability of periodic travling waves for
Korteweg-de Vries type equations with fractional dispersion and other nonlinear
dispersive equations. We establish that a constrained minimizer for the related
variational problem is nonlinearly stable to period preserving perturbations.
We also discuss when the associated linearized equation admits exponentially
growing solutions. The proof utilizes the variational structure of the
equation.
