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.
Highlights
The Camassa-Holm (CH) equation as introduced in [2] is a one dimensional PDE, which is a nonlinear shallow water wave equation [4] and is usually written as (1.1)∂tu − ∂txxu + 3∂xu u − 2∂xxu ∂xu − ∂xxxu u = 0 .This equation has generated a large volume of literature studying its many properties and it has drawn a lot of interest in various communities
The physical relevance of this equation is not limited to shallow water dynamics since, for instance, it has been retrieved as a model for the propagation of nonlinear waves in cylindrical hyper-elastic rods [5]
Consider the group of diffeomorphism Diff(M ) endowed with the right-invariant Hdiv metric, that is, the norm is defined in Eulerian coordinates on the velocity field u by
Summary
The Camassa-Holm (CH) equation as introduced in [2] is a one dimensional PDE, which is a nonlinear shallow water wave equation [4] and is usually written as (1.1). The difficulty is pushed in the fact that the Riemannian manifold on which the equations live is often curved and non-complete, see the end of Section 3.3 This link is interesting from the point of view of classification of fluid dynamic models.
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