Geometric aspects of two- and threepeakons

TL;DR

This paper uses geometric methods to analyze the dynamics of two- and three-peakon solutions of the Camassa-Holm equation, providing new proofs of their asymptotic behavior and computing relevant curvatures.

Contribution

It introduces geometric tools to study peakon dynamics, offering new proofs and curvature computations for two- and three-peakon solutions.

Findings

Recovered known collision conditions

Computed Gauss and sectional curvatures

Provided new geometric proofs of asymptotic behavior

Abstract

We apply geometric tools to study dynamics of two- and threepeakon solutions of the Camassa--Holm equation. New proofs of asymptotic behavior of the solutions are given. In particular we recover well-known collision conditions. Additionally the Gauss curvature (in the twopeakon case) and the sectional curvature (in the treepeakon case) of corresponding manifolds are computed.