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On the Bi-Hamiltonian Theory for the Harry Dym Equation

Marco Pedroni·Vincenzo Sciacca·Jorge P. Zubelli
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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.

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Marco Pedroni, Vincenzo Sciacca, Jorge P. Zubelli. 2001-12-17. On the Bi-Hamiltonian Theory for the Harry Dym Equation. https://arxiv.org/abs/math-ph/0112035

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