Embedding Camassa-Holm equations in incompressible Euler

TL;DR

This paper demonstrates how the Camassa-Holm equations can be embedded within the incompressible Euler equations by representing them as geodesic flows on specific Riemannian manifolds, revealing deep geometric connections.

Contribution

It introduces a novel geometric embedding of the CH2 equations into the incompressible Euler framework via the Hdiv metric in 2D.

Findings

CH2 equations embedded in 2D Euler via Hdiv metric

Euler equations with potential terms embedded in manifold setting

CH2 as a special case of fluid dynamic equations

Abstract

In this article, we show how to embed the so-called CH2 equations into the geodesic flow of the Hdiv metric in 2D, which, itself, can be embedded in the incompressible Euler equation of a non compact Riemannian manifold. The method consists in embedding the incompressible Euler equation with a potential term coming from classical mechanics into incompressible Euler of a manifold and seeing the CH2 equation as a particular case of such fluid dynamic equation.