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Determination of the growth rate for the linearized Zakharov—Kuznetsov equation

Published online by Cambridge University Press:  13 March 2009

M. A. Allen  and
G. Rowlands
M. A. Allen
Affiliation:
Department of Physics, University of Warwick, Coventry CV4 7AL, U.K.
G. Rowlands
Affiliation:
Department of Physics, University of Warwick, Coventry CV4 7AL, U.K.
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Abstract

Studies of the Zakharov—Kuznetsov equation governing solitons in a strongly magnetized ion-acoustic plasma indicate that a perturbed flat soliton is unstable and evolves into higher-dimensional solitons. The growth rate γ = γ(k) of a small sinusoidal perturbation of wavenumber k to a flat soliton has already been found numerically, and lengthy analytical work has given the value of We introduce a more direct analytical method in the form of an extension to the usual multiple-scale perturbation approach and use it to determine a consistent expansion of γ about k = 0 and the other zero at k2 = 5.By combining these results in the form of a two-point Padé approximant, we obtain an analytical expression for γ valid over the entire range of k for which the solution is unstable. We also present a very efficient numerical method for determining the growth rate curve to great accuracy. The Padé approximant gives excellent agreement with the numerical results.

Type
Research Article
Information
Journal of Plasma Physics , Volume 50 , Issue 3 , December 1993 , pp. 413 - 424
Copyright
Copyright © Cambridge University Press 1993

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