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Stability of solitary-wave solutions to a modified Zakharov–Kuznetsov equation

Published online by Cambridge University Press:  29 May 2001

S. MUNRO
Affiliation:
Department of Mathematics, University of Strathclyde, Glasgow G1 1XH, U.K.
E. J. PARKES
Affiliation:
Department of Mathematics, University of Strathclyde, Glasgow G1 1XH, U.K.

Extract

In the context of ion-acoustic waves in a magnetized plasma comprising cold ions and non-isothermal electrons, small-amplitude, weakly nonlinear waves have been shown previously by Munro and Parkes to be governed by a modified version of the Zakharov–Kuznetsov equation. In this paper, we consider solitary travelling-wave solutions to this equation that propagate along the magnetic field. We investigate the initial growth rate γ(k) of a small transverse sinusoidal perturbation of wavenumber k. The instability range is shown to be 0 < k < 3. We use the multiple-scale perturbation method developed by Allen and Rowlands to determine a consistent expansion of γ about k = 0 and k = 3. By combining these results in the form of a Padé approximant, an analytical expression for γ is found that is valid for 0 < k < 3. γ is also determined by using the variational method developed by Bettinson and Rowlands. The two results for γ are compared with a numerical determination.

Type
Research Article
Copyright
© 2000 Cambridge University Press

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