Self-focusing of nonlinear ion-acoustic waves and solitons in magnetized plasmas

The Zakharov-Kuznetsov equation describing Korteweg-de Vries waves and solitons in a strong, uniform magnetic field is rederived taking space stretching to be isotropic. This equation is then used to investigate nonlinear waves and solitons for long-wave instabilities. A solid angle of instability develops around the plane perpendicular to the magnetic field. For weakly nonlinear waves this angle is very narrow: widening as the amplitude of the nonlinear wave is increased. The soliton wave is unstable for all directions other than parallel to the field. Previous results of other authors, limited to solitons and perpendicular propagation are recovered. Calculations are illustrated by polar diagrams for the perturbations. Some broader implications are pointed out.