Transverse stability of plane solitons using the variational method

Abstract

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.