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

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

2004-07-26

arxiv.org/abs/nlin/0407057v2

Nonlinear Sciences

Exactly Solvable and Integrable Systems

11 pages, LaTeX; minor changes, to appear in Int. Math. Res. Not

Scientific paper

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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