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Transverse stability of plane solitons using the variational method

Published online by Cambridge University Press:  01 April 1998

DAVID C. BETTINSON  and
GEORGE ROWLANDS
DAVID C. BETTINSON
Affiliation:
Department of Physics, University of Warwick, Coventry, CV4 7AL, UK
GEORGE ROWLANDS
Affiliation:
Department of Physics, University of Warwick, Coventry, CV4 7AL, UK
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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.

Type
Research Article
Information
Journal of Plasma Physics , Volume 59 , Issue 3 , April 1998 , pp. 543 - 554
Copyright
1998 Cambridge University Press

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