Stability of 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, 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 [gamma](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 [gamma] about k = 0 and k = 3. By combining these results in the form of a Padæ#169; approximant, an analytical expression for [gamma] is found that is valid for 0 < k < 3. [gamma] is also determined by using the variational method developed by Bettinson and Rowlands. The two results for [gamma] are compared with a numerical determination.