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

Stability of Small Periodic Waves in Fractional KdV Type Equations