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

TL;DR

This paper investigates the transverse stability of smooth solitary waves in a two-dimensional generalization of the Camassa-Holm equation, showing they behave similarly to KP-II theory and are linearly stable under small transverse perturbations.

Contribution

It demonstrates the transverse stability of solitary waves in a 2D Camassa-Holm equation and analyzes the spectral behavior of perturbations, extending stability results.

Findings

Double eigenvalue breaks into stable resonances under perturbation

Small-amplitude solitary waves are linearly stable transversely

Transverse perturbations behave similarly to KP-II theory

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.