Stability of KdV solitons with respect to transverse perturbations: Absolute and convective instabilities

We study the stability of one-dimensional solitons propagating in an anisotropic medium. We derived the Kadomtsev-Petviashvili equation for nonlinear waves propagating in an anisotropic medium. By a proper variable substitution this equation reduces either to the KPI or to the KPII equation. In the former case solitons are unstable with respect to the normal modes of transverse perturbations, and in the latter they are stable. We only consider the case when the solitons are unstable. We formulated the linear stability problem. Using the Laplace–Fourier transform, we found the solution describing the evolution of an initial perturbation. Then, using Briggs' method we studied the absolute and convective instabilities. We found that a soliton is convectively unstable unless it propagates at an angle smaller then critical with respect to a critical direction defined by the condition that the group velocity is parallel to the phase velocity. The critical angle is proportional to the ratio of the dispersion length to the soliton width, which is a small parameter. The coefficient of proportionality is expressed in terms of the phase speed and its second derivative with respect to the angle between the propagation direction and the critical direction. As an example we consider the stability of solitons propagating in Hall plasmas.