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 PDEs 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 L2( R) 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.