Integrable geodesic flows on the (Super)extension of the Bott-Virasoro group

In this Letter, we present an answer to the question posed by Marcel, Ovsienko and Roger in their paper (Lett. Math. Phys. 40 (1997), 31-39). The Ito equation, modified dispersive water wave equation and modified dispersionless long wave equation are shown to be the geodesic flows with respect to an L-2 metric on the semidirect product space Diff(s) (S-1) . C-infinity(S-1), where Diff(s)(S-1) is the group of orientation-preserving Sobolev H-s diffeomorphisms of the circle. We also study the geodesic flows with respect to H-1 metric. The geodesic flows in this case yield different integrable systems admitting nonlinear dispersion terms. These systems exhibit more general wave phenomena than usual integrable systems. Finally, we study an integrable geodesic flow on the extended Neveu-Schwarz space.