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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  

Preprint Link:  https://arxiv.org/abs/2402.08669v1 (pdf) 
Date submitted:  20240214 17:33 
Submitted by:  Bonnemain, Thibault 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
Generalised hydrodynamics (GHD) is a recent and powerful framework to study manybody 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 nontrivial. 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) KadomtsevPetviashvili (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).
Current status:
Reports on this Submission
Strengths
1. Provides a novel and synergetic link between generalized hydrodynamics and soliton gas theories for a canonical bidirectional nonlinear wave model.
2. Opens a pathway for the construction of twodimensional soliton gas theory via the connection between the Boussinesq and the KadomtsevPetviashvili equations.
Weaknesses
1. Uses some phenomenological assumptions that are not rigorously proved (e.g. soliton resolution of a random, rapidly decaying Boussinesq wave field in the "timeofflight" construction).
Report
The manuscript is concerned with the construction of generalized hydrodynamics (GHD) of soliton gas for the integrable “twoway” Boussinesq equation. Specifically, the authors extend the previous construction of unidirectional soliton gas GHD for the KdV equation to the (bidirectional) Boussinesq equation. This is an important extension in several respects. The main motivation is the connection of the Boussinesq equation with the KadomtsevPetviashvili (KP) equation, a universal integrable model for weakly twodimensional nonlinear waves. Namely, the Boussinesq equation represents a stationary (2+0) reduction of the KP equation so the GHD of the Boussinesq soliton gas could provide important insights into the behavior of the general KP soliton gas. At the same time, the GHD of the Boussinesq soliton gas is important on its own as the Boussinesq equation is a canonical integrable model exhibiting anisotropic soliton interactions, with the headon and overtaking collision phase shifts having different signs with important consequences for the macroscopic observables in the soliton gas. Generally, the phenomenology of soliton interactions in the Boussinesq equation is much richer than that for the KdV equation. As a result, the GHD generalization of the “isotropic” KdV gas to the anisotropic Boussinesq case is quite nontrivial and exhibits a number of qualitatively new features. In addition to the hydrodynamics of the Boussinesq solitons gas, the authors develop its thermodynamics (entropy, free energy, temperature, etc.) and compute correlations, not available via the standard soliton gas machinery.
The paper is written very well, offering a thorough introduction to the properties of multisoliton solutions of the Boussinesq equation and outlining important parallels between the Boussinesq soliton gas GHD (TBA, GGE, dressing operation, etc.) and the key quantities and relations of its spectral/kinetic theory (density of states, spectral scaling function, nonlinear dispersion relations etc). Overall, this paper represents a valuable contribution to the rapidly growing literature on soliton gas that will have impact on two major communities in modern mathematical and theoretical physics.
I believe the paper meets the SciPost publication criteria and have no hesitation recommending its publication in the essentially present form subject to minor edits suggested below.
Requested changes
1. P.5. Second line after Eq. 5: “Indeed, one can see by direct substitution that the Lax equation is equivalent to the original equation (1)”. This should be augmented by the isospectrality condition, e.g. “provided the spectrum of the operator $L$ is timeinvariant”.
2. P. 5, the line before Eq. (6), replace “conserved charges” by “conserved charges (densities)”.
3. After Eq. (10), replace $t=\infty$ and $x = \infty$ with $t \to \infty$ and $x \to \infty$.
4. The line before Eq. (14): “phase shift” should be “phase shifts”.
5. P.6, the last sentence. I suggest presenting explicit expressions for the KdV phase shifts or simply making a reference to the relevant expressions in Section 4.2
6. P.9 section 2.6: “resonnant” should be “resonant”.
7. P.10. Section 3, the introductory paragraph. “litteral” should be “literal”.
8. P.12, 2 lines before Eq. (40); “Equations (38) is akin…” should be “equations (38) are akin…”.
9. Make a reference to Ref. [31] when introducing the spectral scaling functions after Eq. (40)
10. P.12. Introduce the acronym NDR’s here as it is used later in the text.
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)
Strengths
The paper provides a detailed study of the soliton gas governed by the integrable Businesq equation. While ingredients for such enterprise are known, the issue how to implement them is a difficult task. Interesting novel points are to exploit directly the multisoliton solution of Hirota and the observation to rewrite the TBA equation as a two component system. The text is very wellwritten. Sufficient background is provided so to follow the deductions
Weaknesses
For other integrable models, beyond the soliton gas, there is also a "radiation" background. In the present case is there such a contribution? If yes, why can it be ignored in a hydrodynamic description?
Report
The submitted paper satisfies the level of SciPost Physics. Publication is recommended, subject to the point raised above.
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)
Strengths
1. Timely topic
2. Stateoftheart methods
3. Potential extension to 2+1 dimensional physics
Weaknesses
1. The novelty of the results and their physical relevance are not clearly articulated.
2. No example results given
3. Notation is confusing in some places
4. Text needs thorough grammatical and stylistic revision
Report
This paper considers the generalised hydrodynamics of the integrable Boussinesq equation. This is an interesting problem as it paves the way to a 2+1dimensional extension of the GHD. The scientific relevance of the results justifies publication; however, the level of the results justifying publication in Scipost Physics is not established. In particular, it is unclear to what extent the work goes beyond a straightforward exercise of applying the GHD to another system. The paper would greatly benefit from some examples of time evolution, showing physical features of the dynamics, or some analysis of particular physical properties of the model. In the present form, I propose transferring the paper to Scipost Physics Core.
Requested changes
1. I found the notations a little confusing in connection with the good/bad Boussinesq equation distinction. In (4), the notation omega, which is normally used for angular frequency, is instead used for the imaginary units times the angular frequency. While notations are indeed arbitrary, they definitely made reading harder, all the time having to remind myself of this convention, which is not even pointed out in the text and which goes contrary to the usual one used in the physics literature.
Additionally, I could only assume that the “large deviation principle” is essentially the same as “saddle point approximation”, at least (33), and the subsequent reasoning certainly indicates so. I propose that the authors make this connection clear, as it would make the paper much more readable for a large part of the audience.
I also find the negative sign of the energy function as given in eqn. (68) strange, as this is hard to interpret as the energy of some quasiparticle excitation in the system, which is expected to be positive to keep the energy bounded from below. While this may be a consistent convention, some clarification would be helpful at this point.
2. The use of tenses is confusing and goes against the usual ones in research papers. For example, the authors use future tense “we shall see”, “we shall develop”, etc., referring to other parts of the paper. Other examples are “we will now drop”, “it will be instead more convenient”, and “we will assume”. In all these places, a simple present is the appropriate one to use. Also, present perfect is appropriate when reviewing previous literature, but in places like “we have considered simultaneously the overtaking…”, or the conclusions, a simple past is appropriate.
There are also typos, such as e.g. “litteral” instead of literal; running a spellchecker should greatly help.
3. Section 4.4 on the Euler hydrodynamics is essentially trivial as it repeats the wellknown reasoning to exchange the conservation laws for density equations in GHD. It would make much more sense to present some example solutions of (90), pointing out some characteristic behaviour, answering questions such as
 Is there any physical novelty compared to other applications of the GHD?
 If yes, how are these connected with some special structure of the Boussinesq equation?
Recommendation
Accept in alternative Journal (see Report)