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

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 $0\,{\le}\,\sin\alpha\,{<}\,\sin\alpha_{\rm c}$, 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.