The geometry of the two-component Camassa-Holm and Degasperis-Procesi equations

TL;DR

This paper explores the geometric structure of two-component generalizations of the Camassa-Holm and Degasperis-Procesi equations, revealing their interpretation as geodesic flows on certain diffeomorphism groups and analyzing their curvature properties.

Contribution

It introduces a geometric framework for these two-component equations, proving local well-posedness and providing explicit curvature calculations.

Findings

Equations are geodesic flows on semidirect product groups.

Established local well-posedness in various function spaces.

Identified large subspaces of positive curvature.

Abstract

We use geometric methods to study two natural two-component generalizations of the periodic Camassa-Holm and Degasperis-Procesi equations. We show that these generalizations can be regarded as geodesic equations on the semidirect product of the diffeomorphism group of the circle $\Diff(S^1)$ with some space of sufficiently smooth functions on the circle. Our goals are to understand the geometric properties of these two-component systems and to prove local well-posedness in various function spaces. Furthermore, we perform some explicit curvature calculations for the two-component Camassa-Holm equation, giving explicit examples of large subspaces of positive curvature.