Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Integrable quadratic structures in peakon models   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.

23
The categories of this paper are: solv-int
25
The categories of this paper are: patt-sol
0
The categories of this paper are: hep-ph
5
The categories of this paper are: gr-qc
30
The categories of this paper are: q-alg
28
The categories of this paper are: alg-geom
35
The categories of this paper are: mtrl-th
33
The categories of this paper are: plasm-ph
17
The categories of this paper are: q-fin
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The categories of this paper are: adap-org
10
The categories of this paper are: q-bio
8
The categories of this paper are: hep-ex
37
The categories of this paper are: supr-con
7
The categories of this paper are: hep-th
13
The categories of this paper are: quant-ph
15
The categories of this paper are: nucl-ex