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

Physica Scripta. 2022. Vol. 97. No. 7. Article 075206.

Ruderman M. C., Petrukhin N., Pelinovsky E.

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.

Keywords: dynamics of solitons and quasi-solutions, динамика солитонов и квази-солитонов