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Pure mathematics, Applied mathematics
A class of equations describing the geodesic flow for a right-invariant metric on the group of diffeomorphisms of Rn is reviewed from the viewpoint of their Lie-Poisson structures. A subclass of these equations is analogous to the Euler equations in hydrodynamics (for n = 3), preserving the volume element of the domain of fluid flow. An example in n = 1 dimension is the Camassa-Holm equation, which is a geodesic flow equation on the group of diffeomorphisms, preserving the H1 metric.
R. Ivanov, Poisson Structures of Equations associated with groups of diffeomorphisms, in: Trends in Differential geometry, Complex analysis and Mathematical Physics (ed. K. Sekigawa et al.), World Scientific, 2009, pp 99 – 108.