On the Bi-Hamiltonian Theory for the Harry Dym Equation

Abstract

We describe how the Harry Dym equation fits into the the bi-Hamiltonian formalism for the Korteweg-de Vries equation and other soliton equations. This is achieved by means of a certain Poisson pencil constructed from two compatible Poisson structures. We obtain an analogue of the Kadomtsev-Petviashivili hierarchy whose reduction leads to the Harry Dym hierarchy. We call such system the HD-KP hierarchy. Then, we construct an infinite system of ordinary differential equations (in infinitely many variables) that is equivalent to the HD-KP hierarchy. Its role is analogous to the one played by the Central System for the Kadomtsev-Petviashivili hierarchy.