Hamiltonian Structures on Coadjoint Orbits of Semidirect Product of $G=Diff_+(S^{1})$ and $C^{\infty}(S^1, {\bf R})$

Abstract

We consider the semidirect product of diffeomorphisms of the circle $D={Diff}_+(S^1)$ and $C^{\infty}(S^{1}, {\bf R})$ functions, classify its coadjoint orbits and prove the integrability of hamiltonian (Generalized Dispersive Water Waves (DWW) and KdV-type) systems related to corresponding Lie algebra centrally extended by Kac-Moody, Virasoro and semidirect product cocycles with arbitrary coefficients.