Classical Toda lattice with domain wall initial conditions

- Pedagogical introduction to the generalized hydrodynamics and tba description of Toda lattices

- In some parts, the notation is heavy and hard to follow
- No heuristic mechanism for the development of the density singularity is discussed
- Lack of ab initio numerical simulation to be compared with the GHD

The authors study transport in the classical Toda lattice, focusing on the famous partitioning protocol where two infinite halves of the system are initialized in two different states (in this case, thermal states) and then let evolve.
The Toda lattice is integrable, hence the authors approach the problem within the framework of generalized hydrodynamics (GHD) and report on a singularity in the profiles of the local particle density, which is their main contribution.
I believe the results are interesting and worth to be communicated, but I think they best appeal to a more specialistic audience, hence ScipostCore is probably more appropriate.
In general, I appreciate the pedagogic and detailed introduction the authors provide to the concepts of GHD and in particular of the Toda lattice, but I think in some parts the manuscript is hard to follow and I would ask my comments are taken into account before suggesting the paper for publication.
I feel that not enough space is given to discussing the singularity (which is the central result of the paper) and appears only at the very end of Section 4. Besides, I have some further technical questions that I ask the authors to address.

1. Is there any physical picture or interpretation that can explain such a singularity? If it is so, I think it would important to explicitly mention it.

2. Which are the conditions for the development of such a singularity? Why was it not observed, for example, in Ref. [19]?

3. Since the natural output of the GHD equations is n (which is linked to rho through Eq. (19)), and n is bounded, I do not see why \nu^{-1} diverges. Can the authors comment more on how the singularity is developed in the equations? Is this feature due to the dressing?

4. The singularity 1/|x| is not integrable, hence the number of particles contained in an interval encompassing such a singularity is infinite. However, on the initial state the particle density is finite, hence the singularity is developed through time evolution. How does it happen? Can the authors maybe provide a hydrodynamic description for it? I think that microscopic simulations of the classical model could further elucidate this point, but I understand it requires some extra work.

5. In the conclusions, the authors comment that this singularity seems to be special for the Toda lattice. Can the authors speculate why is it so? What does make Toda special compared with other models? I understand this could be a difficult question, but if the authors have some further insight it would be important to share it.

6. In Eq. (3.8) rho and nu appear for the first time and the authors refer to the preexisting literature for their definition, but maybe some extra comments could make the paper more self-contained. For example, the authors explicitly mention rho to be the particle density, but do not say anything about nu, which is defined only later in Eq. (3.16). Also, if \nu^{-1} is defined as the integral of the particle density, I would expect it to be a positive quantity, while in Fig. 3.c \nu passes through zero and changes sign. Could it be there is a typo in Eq. (3.16) and there is an absolute value missing? I think the absolute value is needed to be consistent with Ref. [17]. Moreover, I would ask the authors to provide an explicit definition of \nu in their manuscript.

7. Lastly, in the abstract it is written “...with a different choice of coordinates smooth behavior is recovered.” and then, at the end of section “Of course, the density really diverges, but when viewed in a different coordinate system the behavior is smooth.” Both sentences sound a bit obscure: which is the coordinate system the authors are referring to? Can they further elaborate on this?

Interesting new phenomenon reported

Clarity of exposition

The paper studies the dynamics in classical Toda lattice from initial domain wall initial conditions.
The framework used for the analysis is that of Generalized Hydrodynamics (GHD).

The focus is on a particular phenomenon observed for a certain class of initial states (thermal states with different temperature, mean velocity and temperature, for specific range of parameter). The phenomenon seems specific to the model under consideration.

While the content is interesting, I think that the clarity of the exposition can be definitely improved.
In "requested changes" I report some example of concepts to be better explained, in order for the paper to be more self-contained and understandable.
After such revision is made, I recommend it for publication is Scipost.

- What is the self-similar solution of GHD? Something is said after eq. 2.14, but should be clearly stated in the paper.
- What is W in eq. (2.13)? I guess it is the Wigner function, but it is not defined
- As far as I understand the paragraph “numerical method” (pag 8) is the algorithm for the equilibrium problem (the initial condition) and not for the GHD equation: I would specify it, since the section is called “GHD of the Toda lattice”
- Simulations with molecular dynamics/hard rods are mentioned more than once, but not properly introduced

1- The paper is well written and presents very well the context of the problem which is analyzed.
2- The subject of the paper is timely and adds new results to a field which is growing, namely, the non-equilibrium dynamics of isolated integrable models, and generalized hydrodynamics in particular.
3- The paper includes a pedagogical introduction to the domain wall problem in section 2.

1- At some points the exposition seems too compressed. Adding more details could help to make the exposition clearer to a broader public.

The authors study the domain wall problem in the Toda chain using generalized hydrodynamics. The most important result is that, for some special parameter values specifying the initial condition, a singularity in the particle density develops a singularity for finite times.
I consider that the paper meets the publication criteria of the journal and recomend publication with minor revisions.

1- Before equation 2.13 it might be useful to explicitly define the linear operator $W$.
2- In order to render the presentation more self-contained and for the sake of clarity, before equation 3.8, provide more details about the GHD equations.
3- The singularity in the non-equilibrium behavior seems to be related to the existence of a critical pressure in equilibrium. Could you elaborate on this relationship?
4- Is it possible to specify the parameter region for which this singularity appears?Is there any identifiable condition on $P_+$ and $P_{-}$ that ensures the appearence of the singularity? As far as I understand, the authors have studied only one point for which the singularity is present.