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.
Ragnisco, O., Bruschi, M. (1996). Peakons, r-matrix and Toda lattice. PHYSICA. A, 228(1-4), 150-159 [10.1016/0378-4371(95)00438-6].
Peakons, r-matrix and Toda lattice
RAGNISCO, Orlando;
1996-01-01
Abstract
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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
