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Euler equations on homogeneous spaces and Virasoro orbits. (English) Zbl 1017.37039
The Korteweg-de Vries $$u_t=-3 u u_x+cu_{xxx},$$ Camassa-Holm $$u_t-u_{txx}=-3 u u_x+2 u_x u_{xxx}+c u_{xxx}$$ and Hunter-Saxon $$u_{txx}=-2 u_x u_{xxx}-u u_{xxx}$$ equations are known to be bihamiltonian and to possess infinitely many conserved quantities as well as soliton and soliton-like solutions. The main goal of the reviewed article is to give the description of these three equations as bihamiltonian systems on the dual to the Virasoro algebra and to relate them to the geometry of the Virasoro coadjoint orbits.
All these equations can be regarded as equations of the geodesic flow associated to different right-invariant metrics on Virasoro group or on appropriate homogeneous spaces. They describe all generic bihamiltonian systems which are related to the Virasoro group and can be integrated by the translation argument principle, as corresponding precisely to the three different types of generic Virasoro orbits. The authors discuss also interrelation between the above metrix and Kähler structures on Virasoro orbits.

##### MSC:
 37K65 Hamiltonian systems on groups of diffeomorphisms and on manifolds of mappings and metrics 53D25 Geodesic flows in symplectic geometry and contact geometry 37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
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