  In this paper, we consider the spectral stability of spatially periodic
traveling wave solutions of the generalized Korteweg-de Vries equation to
long-wavelength perturbations. Specifically, we extend the work of Bronski and
Johnson by demonstrating that the homogenized system describing the mean
behavior of a slow modulation (WKB) approximation of the solution correctly
describes the linearized dispersion relation near zero frequency of the
linearized equations about the background periodic wave. The latter has been
shown by rigorous Evans function techniques to control the spectral stability
near the origin, i.e. stability to slow modulations of the underlying solution.
In particular, through our derivation of the WKB approximation we generalize
the modulation expansion of Whitham for the KdV to a more general class of
equations which admit periodic waves with nonzero mean. As a consequence, we
will show that, assuming a particular non-degeneracy condition, spectral
stability near the origin is equivalent with the local well-posedness of the
Whitham system.
