We present stability results for plane soliton solutions of two
versions of the
two-dimensional KdV equation, namely the Zakharov–Kuznetsov (ZK)
equation and
the Kadomtsev–Petviashvili equation for positive dispersion (KP+
equation). To
do this we use a linear variation-of-action method (VAM). Others have used
this
method, but with little success when applied to these two equations. The
best
results have given the correct instability range, but the predicted growth
rates have
significant errors. For the ZK equation we show, by paying more attention
to the
spatially asymptotic form of the trial function, how better estimates of
the
dispersion relation can be obtained. We go on to obtain the exact dispersion relation for
perturbations of the plane soliton solution of the KP+ equation.