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Integrable quadratic structures in peakon models

by J. Avan, L. Frappat, E. Ragoucy

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Submission summary

Authors (as registered SciPost users): Luc FRAPPAT
Submission information
Preprint Link: scipost_202203_00046v2  (pdf)
Date accepted: 2022-06-30
Date submitted: 2022-05-30 17:58
Submitted by: FRAPPAT, Luc
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Mathematical Physics
Approach: Theoretical


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.

List of changes

Dear Editor,

We have revised our manuscript according to the requested changes of the referees. Here are the major modifications:

- page 3 [section Introduction]
a/ we specified the domain of validity of our study regarding the problem of the boundaries of the Weyl chambers raised by question Q1 of referee 2.
b/ as suggested by the second referee, we have fixed the notation for the matrices appearing in the different models (introducing prime and double prime notation), see comment 1 of referee 2.

- page 5 [section 2.1 Linear Poisson structure]
a/ we made clear the validity of the identification with the r-matrix of Toda molecule after ref. 22, see question Q2 of referee 2.
b/ we explained quickly the notations and conventions for the Lie algebra elements h_i, e_\alpha, etc. after eq. (2.13), see comment 2 of referee 2.

- page 6
The first paragraph of page 6 has been added following the remark 1 of referee 1 about compatible PB on the associative algebra of arbitrary real symmetric matrices ("The Poisson bracket structure (2.11) is indeed ...).

- page 12
Last paragraph before section 4.3 : we commented on the degeneracy of the PB structure obtained in (4.15), see comment 3 of referee 2.

- page 13 [section 5]
In order to comment on remark 4 of referee 1, we expanded the beginning of section 5 (until formula 5.4 included), by pointing out the existence of two objects in the general construction of Poisson commuting Hamiltonians from quadratic structures.
A sentence has also been added after eq. (6.2) on page 15.

Moreover, remark 5.1 on page 14 has been rephrased to clarify the point.

- page 15 [section 6]
we answer the comment 2 of referee 1 on the sign of p_i in relation to the square roots in the Lax matrix.

- page 18 [conclusion]
The conclusion has been expanded with a new paragraph (the first one) by commenting on the newly constructed integrable Hamiltonians, see e.g. comment 4 of referee 1.

Finally, we corrected the typos and done the minors corrections raised by the referees.

Best regards,
The authors

Published as SciPost Phys. 13, 044 (2022)

Reports on this Submission

Report 2 by Laszlo Feher on 2022-6-16 (Invited Report)

  • Cite as: Laszlo Feher, Report on arXiv:scipost_202203_00046v2, delivered 2022-06-16, doi: 10.21468/SciPost.Report.5249


In this revised version the authors have answered the questions and comments that I made in my report on the original manuscript. By this and other minor corrections and additions they have improved the quality of the paper. Thus I have no hesitation to recommend its publication in SciPost Physics.

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Anonymous Report 1 on 2022-6-14 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202203_00046v2, delivered 2022-06-14, doi: 10.21468/SciPost.Report.5233


This is the second version of the paper, which deals with integrable Poisson structures in peakon dynamics. The authors have answered the questions and implemented the suggestions and corrections raised in my report on the first version. As mentioned in the first report, the paper successfully relates the study of peakons with the standard algebraic methods of integrability, such as quadratic integrable Poisson structures, and opens interesting perspectives, for instance on the quantisation of these models. Taking that into account, and with the corrections implemented in this version 2, I am happy to recommend this article for publication in SciPost Physics.

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