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The derivation of a modified Zakharov–Kuznetsov equation and the stability of its solutions

The Zakharov-Kuznetsov equation governs the behaviour of weakly non-linear 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, we show that the reductive perturbation procedure leads to a modified Zakharov–Kuznetsov (mZK) equation. The stability of plane-periodic and solitary travelling-wave solutions of the mZK equation to two-dimensional long-wavelength perturbations is investigated using the method of Rowlands and Infeld.