The stability of solitary wave solutions to a modi

The Zakharov-Kuznetsov equation governs the behaviour of weakly nonlinear ion-acoustic waves in a plasma comprising cold ions and hot isothermal electrons in the presence of a uniform magnetic eld. We consider the more realistic situation in which the electrons are non-isothermal. With an appropriate modiied form of the electron number density proposed by Schamel (1973), we show that the reductive perturbation procedure leads to a modiied Zakharov-Kuznetsov equation. In suitable non-dimensionalised variables and with a convenient scaling the equation is 16u t + 30u 1=2 u x + r 2 u x = 0; where the magnetic eld is in the x-direction. A possible solution to the equation is the plane solitary wave u = sech 4 (x?t) that propagates along the magnetic eld. We show that this solution is unstable to small transverse sinusoidal perturbations of wavenumber k such that 0 < k < 3. Following the method of Allen & Rowlands (1993) we use a multiple-scale perturbation method to determine consistent expansions for the growth rate about k = 0 and k = 3 respectively. By considering a two-point Pad e approximant, we obtain an analytical expression for valid for 0 k 3. We also calculate numerically. The Pad e approximant is in very good agreement with the numerical result.