Integrable quadratic structures in peakon models

Abstract

We propose realizations of the Poisson structures for the Lax representations of three integrable n-body peakon equations, Camassa-Holm, Degasperis-Procesi and Novikov. The Poisson structures derived from the integrability structures of the continuous equations yield quadratic forms for the r-matrix representation, with the Toda molecule classical r-matrix playing a prominent role. We look for a linear form for the r-matrix representation. Aside from the Camassa-Holm case, where the structure is already known, the two other cases do not allow such a presentation, with the noticeable exception of the Novikov model at n=2. Generalized Hamiltonians obtained from the canonical Sklyanin trace formula for quadratic structures are derived in the three cases.