The stability of obliquely-propagating solitary-wave solutions to a modified Zakharov-Kuznetsov equation

Munro, S. and Parkes, E.J. (2004) The stability of obliquely-propagating solitary-wave solutions to a modified Zakharov-Kuznetsov equation. Journal of Plasma Physics, 70 (5). pp. 543-552. (http://dx.doi.org/10.1017/S0022377803002769)

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Abstract

In the context of ion-acoustic waves in a magnetized plasma comprising cold ions and non-isothermal electrons, the present authors have previously shown small amplitude, weakly nonlinear waves to be governed by a modified version of the Zakharov-Kuznetsov equation. In this paper, we consider a plane solitary travelling-wave solution to this equation that propagates at an angle $alpha$ to the magnetic field, where $0,{le},alpha,{le},pi$. The multiple-scale perturbation method developed by Allen and Rowlands is used to calculate the growth rate of a small, transverse, long-wavelength perturbation. To first order there is instability for $0,{le},sinalpha,{<},sinalpha_{ m c}$, where the critical angle $alpha_{ m c}$ is identified. At second order, the singularity which apparently occurs in the growth rate at $alpha,{=},alpha_{ m c}$ is removed by using a method devised by Allen and Rowlands; then it is found that there is also instability for $sinalpha,{ge},sinalpha_{ m c}$. A numerical determination for the growth rate is given for the instability range $0,{<},k,{<},3$, where $k$ is the wavenumber of the perturbation. For $k|{ m sec},alpha|,{ll},1$, there is excellent agreement between the analytical and numerical results. The results in this paper agree qualitatively with those of Allen and Rowlands for the Zakharov-Kuznetsov equation.

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