Peakons, R-Matrix and Toda-Lattice

Abstract

The integrability of a family of hamiltonian systems, describing in a particular case the motionof N ``peakons" (special solutions of the so-called Camassa-Holm equation) is established in the framework of the $r$-matrix approach, starting from its Lax representation. In the general case, the $r$-matrix is a dynamical one and has an interesting though complicated structure. However, for a particular choice of the relevant parameters in the hamiltonian (the one corresponding to the pure ``peakons" case), the $r$-matrix becomes essentially constant, and reduces to the one pertaining to the finite (non-periodic) Toda lattice. Intriguing consequences of such property are discussed and an integrable time discretisation is derived.

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References in corpus (2)

• An integrable shallow water equation with peaked solitons
• Integrable Time-Discretisation of the Ruijsenaars-Schneider Model

Cited by in corpus (16)

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• Long time behaviour for a class of low-regularity solutions of the Camassa-Holm equation
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• Projective reduction of the discrete Painlevé system of type $(A_2+A_1{)}^{(1)}$
• Integrable discretizations of the spin Ruijsenaars-Schneider models
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• Isospectral flows related to Frobenius-Stickelberger-Thiele polynomials
• The Camassa-Holm hierarchy, related N-dimensional integrable systems, and algebro-geometric solution on a symplectic submanifold
• A new integrable structure associated to the Camassa-Holm peakons
• R-matrix for a geodesic flow associated with a new integrable peakon equation
• On Calogero-Francoise-type Lax matrices and their dynamical r-matrices