Geodesic Flow on Extended Bott-Virasoro Group and Generalized Two Component Peakon Type Dual Systems

This paper discusses several algorithmic ways 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(S)⋉ C∞(S) 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 OlverRosenau’s paper [31]. Mathematics Subject Classifications (2000): 53A07, 53B50.