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

Westudy the stability of one-dimensional solitons propagating in an anisotropic medium.Wederived

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.Weonly consider the case when the solitons are unstable.Weformulated 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.Wefound 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.