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 sl(2,R). The reduction process is a bi-Hamiltonian reduction that can be canonically performed on every bi-Hamiltonian manifold.
Lorenzoni, P., Pedroni, M. (2004). On the bi-Hamiltonian structures of the Camassa-Holm and Harry Dym equations. INTERNATIONAL MATHEMATICS RESEARCH NOTICES(75), 4019-4029 [10.1155/S1073792804142554].
On the bi-Hamiltonian structures of the Camassa-Holm and Harry Dym equations
LORENZONI, PAOLO;
2004
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 sl(2,R). The reduction process is a bi-Hamiltonian reduction that can be canonically performed on every bi-Hamiltonian manifold.
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