On the stability of periodic solutions of the generalized Benjamin-Bona-Mahony equation, submitted

Abstract. We study the stability of a four parameter family of spatially periodic trav-eling wave solutions of the generalized Benjamin-Bona-Mahony equation to two classes of perturbations: periodic perturbations with the same periodic structure as the underly-ing wave, and long-wavelength localized perturbations. In particular, we derive necessary conditions for spectral instability to perturbations to both classes of perturbations by de-riving appropriate asymptotic expansions of the periodic Evans function, and we outline a nonlinear stability theory to periodic perturbations based on variational methods which effectively extends our periodic spectral stability results. 1