Stability of Small Periodic Waves in Fractional KdV Type Equations

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.