Relation between the Negative-Order Harry Dym Hierarchy and a Family of Backward Neumann Type Systems

Abstract

From the bidirectional Lenard gradients, the negative-order Harry Dym (nHD) hierarchy is retrieved and further embedded into a bi-Hamiltonian structure displaying integrability. It follows from Neumann type integrable reduction that the nHD hierarchy is reduced to a family of backward Neumann type systems, which separate the temporal and spatial variables on the tangent bundle of an ellipsoid. Backward Neumann type systems are then proved to be completely integrable in the Liouville sense. From the commutativity of backward Neumann type flows, the relation between the nHD hierarchy and backward Neumann type systems is specified, where the involutive solutions of backward Neumann type systems yield the finite parametric solutions of the nHD hierarchy. Moreover, we propose the concept of a negative-order Novikov equation that cuts out a finite-dimensional invariant subspace for a negative-order integrable system, which paves an alternative way to obtain explicit solutions of negative-order integrable nonlin...