Geodesic flow on extended bott-virasoro group and generalized two-component peakon type dual systems

This paper discusses an algorithmic way of constructing integrable evolution equations based on Lie algebraic structure. We derive, in a pedagogical style, a large class of two-component peakon type dual systems from their two-component soliton equations counter part. We study the essential aspects of Hamiltonian flows on coadjoint orbits of the centrally extended semidirect product group Diff(S¹) ⋉ C^∞(S ¹) to give a systematic derivation of the dual counter parts of various two-component of integrable systems, viz., the dispersive water wave equation, the Kaup-Boussinesq system and the Broer-Kaup system, using moment of inertia operators method and the (frozen) Lie-Poisson structure. This paper essentially gives Lie algebraic explanation of Olver-Rosenau's paper [31].