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

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 field. We consider the more realistic situation in which the electrons are non-isothermal. With an appropriate modified form of the electron number density proposed by Schamel (1973), we show that the reductive perturbation procedure leads to a modified 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 field is in the x-direction. A possible solution to the equation is the plane solitary wave u = sech 4 (x \Gamma t) that propagates along the magnetic field. 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 de..