We study the stability of a four parameter family of spatially periodic
traveling wave solutions of the generalized Benjamin-Bona-Mahony equation to
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 to
perturbations to both classes of perturbations by deriving 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.Comment: 27 pages, 3 figure