Weakly nonlinear waves in magnetized plasma with a slightly non-Maxwellian electron distribution. Part 1. Stability of solitary waves

Weakly nonlinear waves in strongly magnetized plasma with slightly non-isothermal electrons are governed by a modified Zakharov-Kuznetsov (ZK) equation, containing both quadratic and half-order nonlinear terms, which we refer to as the Schamel-Korteweg-de Vries-Zakharov-Kuznetsov (SKdVZK) equation. We present a, method to obtain an approximation for the growth rate, gamma, of sinusoidal perpendicular perturbations of wavenumber, k. to SKdVZK solitary waves over the entire range of instability. Unlike for (modified) ZK equations with one nonlinear term, in this method there is no analytical expression for k(c), the cut-off wavenumber (at which the growth rate is zero) or its corresponding eigenfunction. We therefore obtain approximate expressions for these using an expansion parameter, a, related to the ratio of the nonlinear terms. The expressions are then Used to find gamma for k near k(c) as a function of a The approximant derived from combining these analytical results with the ones for small k agrees very well with the values of gamma obtained numerically. It is found that both k(c) and the maximum growth rate decrease as the electron distribution becomes progressively less peaked than the Maxwellian. We also present new algebraic and rarefactive solitary wave solutions to the equation.