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A new integrable structure associated to the Camassa-Holm peakons
by J. Avan, L. Frappat, E. Ragoucy
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Submission summary
| Authors (as registered SciPost users): | Eric Ragoucy |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202310_00041v1 (pdf) |
| Date submitted: | 2023-10-31 13:55 |
| Submitted by: | Ragoucy, Eric |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We provide a closed Poisson algebra involving the Ragnisco--Bruschi generalization of peakon dynamics in the Camassa--Holm shallow-water equation. This algebra is generated by three independent matrices. From this presentation, we propose a one-parameter integrable extension of their structure. It leads to a new $N$-body peakon solution to the Camassa--Holm shallow-water equation depending on two parameters. We present two explicit constructions of a (non-dynamical) $r$-matrix formulation for this new Poisson algebra. The first one relies on a tensorization of the $N$-dimensional auxiliary space by a 4-dimensional space. We identify a family of Poisson commuting quantities in this framework, including the original ones. This leads us to constructing a second formulation identified as a spectral parameter representation.
Author comments upon resubmission
We thank the referees for their careful reading of our manuscript. We have revised our manuscript according to the requested changes of the referees (see details below).
Yours sincerely
The authors
List of changes
Here are the major modifications:
- point 1 (the main point): there was indeed a sign mistake in the coefficient c'. Therefore, we were able to exhibit a spectral parameter dependent, non dynamical r-matrix. Accordingly, the rest of the section after eq. (4.19) has been completely rewritten, making explicit the form of the r-matrix. We also propose a parametrization of the complex variety on which lives the spectral parameter (4.21 and 4.22).
- point 2: before (2.9) a relic sentence with reference to an unlabelled appendix has been removed.
- point 3: a missing factor 1/n has been added in (old) eq. (2.24), now (2.25).
- point 4: after eq. (4.15), we made explicit different choices for the value x, and recover the initial matrix for x=1, as pointed out by the referee.
- point 5: adequate \nu factor added in (4.17) and ff.
- point 6: tensor index 2 has been removed.
- point 7: details of the properties used in the calculation of eq. (2.29) have been added in section 2.3 (between 2.22 and 2.29).
- point 8: comment on the choice (2.32). First, we have slightly reorganized section 2.4 with paragraph subsections. Then, we have added a (unnumbered) subsection to explain how to obtain the case v generic from the cases v^2=1.
We added a sentence in the acknowledgements to thank the referees for their suggestions leading to a substantial improvement of section 4.3.
Best regards,
The authors
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 4) on 2023-11-14 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202310_00041v1, delivered 2023-11-14, doi: 10.21468/SciPost.Report.8112
Report
The authors have implemented relevant modifications on the manuscript following the reports on the first version. In particular, they now discuss the r-matrix formulation with spectral parameter and added a useful parametrisation of the latter in terms of hyperbolic functions.
In my opinion, this paper is clearly written and presents interesting new results on the integrable structure of peakon-type solutions of the Camassa-Holm equation. I am therefore happy to recommend it for publication in SciPost Physics.
I indicate in the "Requested changes" section a few points which I think are typos. They should be checked by the authors and, if needed, corrected in the final version of the manuscript. However, these are minors points and in my opinion they do not require another refereeing step before publishing.
Requested changes
1- Mainly, it seems to me that there is a typo in the expression (4.22) of the r-matrix in terms of hyperbolic functions, changing a sign and some "cosh" into "sinh":
\[ r_{12}(z_1,z_2) = - \Gamma_{12} + \frac{\cosh(z_1)+\cosh(z_2)}{\sinh(z_1)-\sinh(z_2) } \left(\alpha \sinh(z_2)-\frac{\gamma}{2}\right)\Pi \]
\[+ \frac{\cosh(z_1)-\cosh(z_2)}{\sinh(z_1)-\sinh(z_2) } \left(\alpha \sinh(z_2)-\frac{\gamma}{2}\right)\Pi^t + \alpha\cosh(z_2) (\Pi-\Pi^t) \]
2- After (2.22), $\text{Tr}_2(\Pi)$ should be $\mathbb{I}_{N}$ instead of $\mathbb{I}_{N^2}$.
3- It seems to me that (2.23) and (2.24) are now the same and that (2.23) should have a term $[\Pi^t,A_2]$, as it is before using the property $[\Pi^t,M_2]=[\Pi^t,M^t_1]$. Moreover, I'm not sure this property was used to derive (2.26) from (2.25), but rather the next step (2.27).
4- Before (2.28), I think there is a $\lambda$ missing in $[\Pi^t,\bar{L}_1-\bar{L}_2] = 2\lambda [\Pi^t,A_1]$.