  We consider the effects of varying dispersion and nonlinearity on the
stability of periodic traveling wave solutions of nonlinear PDE of KdV-type,
including generalized KdV and Benjamin-Ono equations. In this investigation, we
consider the spectral stability of such solutions that arise as small
perturbations of an equilibrium state. A key feature of our analysis is the
development of a nonlocal Floquet-like theory that is suitable to analyze the
$L^2(\RM)$ spectrum of the associated linearized operators. Using spectral
perturbation theory then, we derive a relationship between the power of the
nonlinearity and the symbol of the fractional dispersive operator that
determines the spectral stability and instability to arbitrary small localized
perturbations.
