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The Camassa-Holm equation as a geodesic flow on the diffeomorphism group

Shinar Kouranbaeva
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Abstract

Misiolek has shown that the Camassa-Holm (CH) equation is a geodesic flow on the Bott-Virasoro group. In this paper it is shown that the Camassa-Holm equation for the case $κ=0$ is the geodesic spray of the weak Riemannian metric on the diffeomorphism group of the line or the circle obtained by right translating the $H^1$ inner product over the entire group. This paper uses the right-trivialisation technique to rigorously verify that the Euler-Poincaré theory for Lie groups can be applied to diffeomorphism groups. The observation made in this paper has led to physically meaningful generalizations of the CH-equation to higher dimensional manifolds (see Refs. \cite{HMR} and \cite{SH}).

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Shinar Kouranbaeva. 1998-07-22. The Camassa-Holm equation as a geodesic flow on the diffeomorphism group. https://doi.org/10.1063/1.532690

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