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

2004/07/26 by Paolo Lorenzoni, Lorenzoni, Paolo, Marco Pedroni +1 · 2 citations
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Differential Geometry (math.DG) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Nonlinear Waves and Solitons #math.DG #nlin.SI

paper · pdf · doi:10.48550/arxiv.nlin/0407057

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

openalex publication_date 2004/07/26 · arxiv created 2004/10/04 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 \mathfraksl(2,ℝ). The reduction process is a bi-Hamiltonian reduction, that can be canonically performed on every bi-Hamiltonian manifold.

Cited by

• Some Remarks on the KP System of the Camassa-Holm Hierarchy
• A new two-component integrable system with peakon solutions

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