Integrable quadratic structures in peakon models

TL;DR

This paper explores the Poisson and r-matrix structures of three integrable peakon equations, revealing quadratic forms and limitations in linear representations, and derives generalized Hamiltonians for these models.

Contribution

It introduces realizations of Poisson structures for peakon equations and investigates their r-matrix representations, highlighting differences among models and deriving generalized Hamiltonians.

Findings

Quadratic Poisson structures are derived for three peakon models.

Linear r-matrix representations are only feasible for the Camassa-Holm and Novikov models at n=2.

Generalized Hamiltonians are obtained from the Sklyanin trace formula.

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.