On a two-component $\pi$-Camassa--Holm system

TL;DR

This paper introduces a new two-component $$-Camassa--Holm system modeled as a geodesic flow on a semidirect product of the circle's diffeomorphism group, providing geometric insights and well-posedness results.

Contribution

It develops a geometric formalism for the $$-Camassa--Holm system, extending previous work on the two-component Camassa--Holm equation with explicit curvature computations.

Findings

Establishes well-posedness of the new system

Provides explicit sectional curvature calculations

Connects the system to geometric flow on Lie groups

Abstract

A novel $\pi$-Camassa--Holm system is studied as a geodesic flow on a semidirect product obtained from the diffeomorphism group of the circle. We present the corresponding details of the geometric formalism for metric Euler equations on infinite-dimensional Lie groups and compare our results to what has already been obtained for the usual two-component Camassa--Holm equation. Our approach results in well-posedness theorems and explicit computations of the sectional curvature.