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

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 {≤} alpha {≤} 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 {≤} sinalpha {<} sinalpha_c, where the critical angle alpha_c is identified. At second order, the singularity which apparently occurs in the growth rate at alpha {=} alpha_c is removed by using a method devised by Allen and Rowlands; then it is found that there is also instability for sinalpha {≥} sinalpha_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|sec alpha| {≪} 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.