Soliton gas of the integrable Boussinesq equation and its generalised hydrodynamics

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 two-dimensional soliton gas theory via the connection between the Boussinesq and the Kadomtsev-Petviashvili equations.

1. Uses some phenomenological assumptions that are not rigorously proved (e.g. soliton resolution of a random, rapidly decaying Boussinesq wave field in the "time-of-flight" construction).

The manuscript is concerned with the construction of generalized hydrodynamics (GHD) of soliton gas for the integrable “two-way” 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 Kadomtsev-Petviashvili (KP) equation, a universal integrable model for weakly two-dimensional 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 head-on 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.

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 time-invariant”.

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.

Publish (easily meets expectations and criteria for this Journal; among top 50%)

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 multi-soliton solution of Hirota and the observation to rewrite the TBA equation as a two component system. The text is very well-written. Sufficient background is provided so to follow the deductions

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?

The submitted paper satisfies the level of SciPost Physics. Publication is recommended, subject to the point raised above.

Publish (easily meets expectations and criteria for this Journal; among top 50%)

1. Timely topic
2. State-of-the-art methods
3. Potential extension to 2+1 dimensional physics

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

This paper considers the generalised hydrodynamics of the integrable Boussinesq equation. This is an interesting problem as it paves the way to a 2+1-dimensional 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.

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 quasi-particle 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 spell-checker should greatly help.

3. Section 4.4 on the Euler hydrodynamics is essentially trivial as it repeats the well-known 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?

Accept in alternative Journal (see Report)