A. Geyer, Y. Liu, and D.E. Pelinovsky

On the transverse stability of smooth solitary waves in a two-dimensional Camassa-Holm equation

Abstract:
We consider the propagation of smooth solitary waves in a two-dimensional generalization of the Camassa–Holm equation. We show that transverse perturbations to one-dimensional solitary waves behave similarly to the KP-II theory. This conclusion follows from our two main results: (i) the double eigenvalue of the linearized equations related to the translational symmetry breaks under a transverse perturbation into a pair of the asymptotically stable resonances and (ii) small-amplitude solitary waves are linearly stable with respect to transverse perturbations.

Keywords:
Camassa-Holm equation; Kadomtsev-Petviashvili equation; smooth solitary waves, existence and transverse stability, resonant poles, asymptotic stability.