The Camassa-Holm equation as a geodesic flow on the diffeomorphism group. (English) Zbl 0958.37060

The paper deals with a physically meaningful generalization of the Camassa-Holm (CH) equation to higher dimension. In particular it is shown that the (CH) equation is the geodesic flow of the weak Riemannian metric on the diffeomorphism group of the line or the circle obtained by right translating in \(H^1\).


37K65 Hamiltonian systems on groups of diffeomorphisms and on manifolds of mappings and metrics
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
58B20 Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds
58D05 Groups of diffeomorphisms and homeomorphisms as manifolds
86A05 Hydrology, hydrography, oceanography
Full Text: DOI arXiv


[1] DOI: 10.1103/PhysRevLett.71.1661 · Zbl 0972.35521
[2] DOI: 10.1016/S0065-2156(08)70254-0
[3] DOI: 10.2307/1970699 · Zbl 0211.57401
[4] DOI: 10.1017/S030821050002477X · Zbl 0795.58018
[5] DOI: 10.1006/aima.1998.1721 · Zbl 0951.37020
[6] DOI: 10.1006/jfan.1998.3335 · Zbl 0933.58010
[7] DOI: 10.5802/aif.233 · Zbl 0148.45301
[8] DOI: 10.1016/S0393-0440(97)00010-7 · Zbl 0901.58022
[9] DOI: 10.1007/BF00739423 · Zbl 0808.35124
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