  We derive the Whitham modulation equations for the Zakharov-Kuznetsov
equation via a multiple scales expansion and averaging two conservation laws
over one oscillation period of its periodic traveling wave solutions. We then
use the Whitham modulation equations to study the transverse stability of the
periodic traveling wave solutions. We find that all such solutions are linearly
unstable, and we obtain an explicit expression for the growth rate of the most
unstable wave numbers. We validate these predictions by linearizing the
equation around its periodic solutions and solving the resulting eigenvalue
problem numerically. Finally, we calculate the growth rate of the solitary
waves analytically. The predictions of Whitham modulation theory are in
excellent agreement with both of these approaches.
