TY  - JOUR
PB  - SciPost Foundation
DO  - 10.21468/SciPostPhys.13.2.044
TI  - Integrable quadratic structures in peakon models
PY  - 2022/08/31
UR  - https://scipost.org/SciPostPhys.13.2.044
JF  - SciPost Physics
JA  - SciPost Phys.
VL  - 13
IS  - 2
SP  - 044
A1  - Avan, Jean
AU  - Frappat, Luc
AU  - Ragoucy, Eric
AB  - 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.
ER  -

@Article{10.21468/SciPostPhys.13.2.044,
	title={{Integrable quadratic structures in peakon models}},
	author={J. Avan and L. Frappat and E. Ragoucy},
	journal={SciPost Phys.},
	volume={13},
	pages={044},
	year={2022},
	publisher={SciPost},
	doi={10.21468/SciPostPhys.13.2.044},
	url={https://scipost.org/10.21468/SciPostPhys.13.2.044},
}