Peakons, r-matrix and Toda lattice

The integrability of a family of Hamiltonian systems, describing in a particular case the motion of 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 this property are discussed and an integrable time discretisation is derived.