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Soliton gas of the integrable Boussinesq equation and its generalised hydrodynamics
by Thibault Bonnemain, Benjamin Doyon
Submission summary
| Authors (as registered SciPost users): | Thibault Bonnemain |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2402.08669v2 (pdf) |
| Date submitted: | 2024-12-06 11:03 |
| Submitted by: | Bonnemain, Thibault |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
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| Approach: | Theoretical |
Abstract
Generalised hydrodynamics (GHD) is a recent and powerful framework to study many-body integrable systems, quantum or classical, out of equilibrium. It has been applied to several models, from the delta Bose gas to the XXZ spin chain, the KdV soliton gas and many more. Yet it has only been applied to (1+1)-dimensional systems and generalisation to higher dimensions of space is non-trivial. We study the Boussinesq equation which, while generally considered to be less physically relevant than the KdV equation, is interesting as a stationary reduction of the (boosted) Kadomtsev-Petviashvili (KP) equation, a prototypical and universal example of a nonlinear integrable PDE in (2+1) dimensions. We follow a heuristic approach inspired by the Thermodynamic Bethe Ansatz in order to construct the GHD of the Boussinesq soliton gas. Such approach allows for a statistical mechanics interpretation of the Boussinesq soliton gas that comes naturally with the GHD picture. This is to be seen as a first step in the construction of the KP soliton gas, yielding insight on some classes of solutions from which we may be able to build an intuition on how to devise a more general theory. This also offers another perspective on the construction of anisotropic bidirectional soliton gases previously introduced phenomenologically by Congy et al (2021).
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Author comments upon resubmission
We thank the referees for the careful reading of the manuscript and the useful comments which helped improve the presentation and make it clearer. We furthermore apologise for the delayed reply, while an updated version of the manuscript was put on arXiv soon after having received the editor's recommendation, both the authors got busy and the final steps of the resubmission were put on standby.
Report #1 by Referee 2
- Notation conventions have been changed when discussing the difference between the Good and Bad Boussinesq equation.
Similarities between the large deviation principle and the saddle point approximation were highlighted.
We added a footnote page 17 to discuss the sign of the energy function. The short of it is that this convention was chosen in order to ensure that the momentum function of right-moving solitons would be positive, while that of left-moving would be negative. Furthermore, the expression of the energy function, combined with the restrictions on the spectral parameter \eta, ensures it is bounded from below.
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We corrected the typos, but did not modify our use of tenses.
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A small paragraph was added at the end of page 20 highlighting some specificities of the Boussinesq phenomenology, as well as making reference to another paper by the authors in which those specificities are discussed further. Said paper also features some examples of solutions to Eq. (90).
Report #2 by Referee 1
A paragraph was added in Section 5, at the end of page 21 to highight the relevance of a gas made of only solitons and why one may forego radiations. The main reason is that, in the large N limit, N-soliton solutions can provide excellent approximations of general solutions (even ones involving radiations) on any compact (x,t) domain (see e.g. refs 9, 19 and the newly cited 108, 109).
Report #3 by Referee 3
All the comments by Referee 3 were taken into account, c.f. the list of changes.
List of changes
Several typos were corrected.
Some further references were added in the introduction.
We changed the notations of Section 2.1 in accordance with the suggestions of Referre 2.
Mention of the iso-spectrality condition was added.
A footnote was added page 6 to clarify the origin of the name "spectral parameter".
We added the explicit expression for the KdV phase shift and made a reference to section 4.2 at the beginning of page 7.
A reference to [31] was added after equation (40), page 12, and the acronym NDR was introduced in the same sentence.
Mention of the link between the large deviation principle and the saddle point approximation was added page 15.
A footnote was added page 17 to discuss the sign of the energy function.
A paragraph was added at the end of page 20 highlighting some specificities of the Boussinesq phenomenology, as well as making reference to another paper by the authors in which those specificities are discussed further.
A paragraph was added in Section 5, at the end of page 21 to highight the relevance of a gas made of only solitons and why one may forego radiations.