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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 | |
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Preprint Link: | scipost_202203_00046v2 (pdf) |
Date accepted: | June 30, 2022 |
Date submitted: | May 30, 2022, 5:58 p.m. |
Submitted by: | FRAPPAT, Luc |
Submitted to: | SciPost Physics |
Ontological classification | |
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Academic field: | Physics |
Specialties: |
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Approach: | Theoretical |
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.
List of changes
Dear Editor,
We have revised our manuscript according to the requested changes of the referees. Here are the major modifications:
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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.
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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.
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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 ...).
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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.
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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.
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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.
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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 (Referee 1) 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
Report
Report #1 by Anonymous (Referee 2) 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