Deformed Harry Dym and Hunter-Zheng Equations

Abstract

We study the deformed Harry Dym and Hunter-Zheng equations with two arbitrary deformation parameters. These reduce to various other known models in appropriate limits. We show that both these systems are bi-Hamiltonian with the same Hamiltonian structures. They are integrable and belong to the same hierarchy corresponding to positive and negative flows. We present the Lax pair description for both the systems and construct the conserved charges of negative order from the Lax operator. For the deformed Harry Dym equation, we construct the non-standard Lax representation for two special classes of values of the deformation parameters. In general, we argue that a non-standard description will involve a pseudo-differential operator of infinite order.

Latex file, 15 pages

References in corpus (3)

• Supersymmetric Extensions of the Harry Dym Hierarchy
• The Generalized Harry Dym Equation
• A singular integrable equation from short capillary-gravity waves

Cited by in corpus (3)

• The Bi-Hamiltonian Structure of the Short Pulse Equation
• On negative flows of the AKNS hierarchy and a class of deformations of bihamiltonian structure of hydrodynamic type
• A vectorial binary Darboux transformation of the first member of the negative part of the AKNS hierarchy