Geodesic flow on (super-) Bott–Virasoro group and Harry Dym family

Abstract

We show various Harry Dym type equations and Super Harry Dym equations, introduced by Brunelli, Das, and Popowicz [2003 Supersymmetric extensions of the Harry Dym hierarchy, J. Math. Phys. 44, 4756–4767 (2003)], follow from the geodesic flows on the Bott–Virasoro group and its supersymmetric generalization, superconformal group. In fact, their bi-Hamiltonian structures can be derived from the Lie Poisson structures on the (super-) Bott–Virasoro orbit. We also show that that m2KdV or Calogero–Degasperis equation is also connected to the Bott–Virasoro group.