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 Zahkarov-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 pade approximant, 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.