Hamiltonian formulation of generalized peakon type systems

Abstract: Here we give a pedagogical introduction to Euler-Poincar\'e flows on the (extended) Virasoro orbit, divided into three parts. In the first part, we introduce some basic notions of Euler-Poincar\'e flows, Virasoro orbit etc. The second part describes an Euler-Poincar\'e formulation of a new class of peakon equations with cubic nonlinearity found by Z. Qiao and V. Novikov. We show that the Hamiltonian structures obtained by Qiao and Hone and Wang can be obtained by our method. The last part introduces integrable ODEs associated to stabilizer set of Virasoro orbit.