On the stability of periodic solutions of the generalized Benjamin–Bona–Mahony equation

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