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On the bi-Hamiltonian structures of the Camassa-Holm and Harry Dym equations

Paolo Lorenzoni·Marco Pedroni
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Abstract

We show that the bi-Hamiltonian structures of the Camassa-Holm and Harry Dym hierarchies can be obtained by applying a reduction process to a simple Poisson pair defined on the loop algebra of $\mathfrak{sl}(2,\mathbb{R})$. The reduction process is a bi-Hamiltonian reduction, that can be canonically performed on every bi-Hamiltonian manifold.

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Paolo Lorenzoni, Marco Pedroni. 2004-10-04. On the bi-Hamiltonian structures of the Camassa-Holm and Harry Dym equations. https://arxiv.org/abs/nlin/0407057

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