Journal of Plasma Physics


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

S. MUNRO a1 and E. J. PARKES a1
a1 Department of Mathematics, University of Strathclyde, Glasgow G1 1XH, UK (

Article author query
munro s   [Google Scholar] 
parkes ej   [Google Scholar] 


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 $0\,{\le}\,\sin\alpha\,{<}\,\sin\alpha_{\rm c}$, where the critical angle $\alpha_{\rm c}$ is identified. At second order, the singularity which apparently occurs in the growth rate at $\alpha\,{=}\,\alpha_{\rm c}$ is removed by using a method devised by Allen and Rowlands; then it is found that there is also instability for $\sin\alpha\,{\ge}\,\sin\alpha_{\rm c}$. 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 $k|{\rm sec}\,\alpha|\,{\ll}\,1$, 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.

(Received September 10 2003)
(Accepted September 18 2003)