  In this paper, we investigate the instability of one-dimensionally stable
periodic traveling wave solutions of the generalized Korteweg-de Vries equation
to long wavelength transverse perturbations in the generalized
Zakharov-Kuznetsov equation in two space dimensions. By deriving appropriate
asymptotic expansions of the periodic Evans function, we derive an index which
yields sufficient conditions for transverse instabilities to occur. This index
is geometric in nature, and applies to any periodic traveling wave profile
under some minor smoothness assumptions on the nonlinearity. We also describe
the analogous theory for periodic traveling waves of the generalized
Benjamin-Bona-Mahony equation to long wavelength transverse perturbations in
the gBBM-Zakharov-Kuznetsov equation.
