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Projective structure and integrable geodesic flows on the extension of Bott-Virasoro group. (English) Zbl 1126.37319

Summary: This is a sequel to our paper [Lett. Math. Phys. 52, No. 4, 311–328 (2000; Zbl 0997.37050)], triggered from a question posed by P. Marcel, V. Ovsienko and C. Roger [Lett. Math. Phys. 40, 31–39 (1997; Zbl 0881.17019)]. In this paper, we show that the multicomponent (or vector) Ito equation, modified dispersive water wave equation, and modified dispersionless long wave equation are the geodesic flows with respect to an \(L^2\) metric on the semidirect product space \(\widehat{\text{Diff}^s(S^1)\ltimes C^{\infty}(S^1)^k}\), where \(\text{Diff}^s(S^1)\) is the group of orientation preserving Sobolev \(H^s\) diffeomorphisms of the circle. We also study the projective structure associated with the matrix Sturm-Liouville operators on the circle.

MSC:

37K65 Hamiltonian systems on groups of diffeomorphisms and on manifolds of mappings and metrics
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37K30 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures
58D05 Groups of diffeomorphisms and homeomorphisms as manifolds
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