Deformed Harry Dym and Hunter-Zheng Equations

Details Deformed Harry Dym and Hunter-Zheng Equations Deformed Harry Dym and Hunter-Zheng Equations

2003-07-23

arxiv.org/abs/nlin/0307043v1

J.Math.Phys.45:2646-2655,2004

Nonlinear Sciences

Exactly Solvable and Integrable Systems

Latex file, 15 pages

Scientific paper

10.1063/1.1756697

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.

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