The three-dimensional stability of solitary shear kinetic Alfvén waves in a low-β plasma is investigated by the method of Zakharov & Rubenchik (1974). It is found that there is no instability if the direction of perturbation falls within a certain region of space. The growth rate of the instability for the unstable region is determined. This growth rate is found to decrease with increasing angle between the direction of propagation of the solitary wave and the direction of the external uniform magnetic field. A particular case of the present analysis gives results on the stability of ion-acoustic solitons in a magnetized plasma.

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Following a method developed in the theory of long water waves, stable approximate equations for ion-acoustic waves are derived from approximate Hamiltonians. Results are given for three cases, viz., low amplitude, long wavelength, and the Boussinesq approximation for fairly long, fairly low waves.

The Zakharov-Kuznetsov equation describes nonlinear ion-acoustic waves in a strongly magnetized plasma and is used here to investigate the stability of nonlinear periodic and solitary ion-acoustic waves in a strong uniform magnetic field with respect to long wavelength plane wave perturbations in an arbitrary direction. A dispersion relation is obtained in the form of a quadratic equation. From this it is found that there is instability when the angle between the direction of propagation and the external magnetic field exceeds a critical value. The growth rate is computed and attains a maximum when the said angle is π/2. A solitary wave is always found to be unstable unless the perturbation is along the direction of propagation.