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Triangular GrossPitaevskii breathers and DamskiChandrasekhar shock waves
by M. Olshanii, D. Deshommes, J. Torrents, M. Gonchenko, V. Dunjko, G. E. Astrakharchik
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Submission summary
As Contributors:  Vanja Dunjko · Maxim Olshanii 
Arxiv Link:  https://arxiv.org/abs/2102.12184v5 (pdf) 
Date submitted:  20210329 08:42 
Submitted by:  Dunjko, Vanja 
Submitted to:  SciPost Physics 
Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
The recently proposed map [arXiv:2011.01415] between the hydrodynamic equations governing the twodimensional triangular coldbosonic breathers [Phys. Rev. X 9, 021035 (2019)] and the highdensity zerotemperature triangular freefermionic clouds, both trapped harmonically, perfectly explains the former phenomenon but leaves uninterpreted the nature of the initial ($t=0$) singularity. This singularity is a density discontinuity that leads, in the bosonic case, to an infinite force at the cloud edge. The map itself becomes invalid at times $t<0$. A similar singularity appears at $t = T/4$, where $T$ is the period of the harmonic trap, with the FermiBose map becoming invalid at $t > T/4$. Here, we first map  using the scale invariance of the problem  the trapped motion to an untrapped one. Then we show that in the new representation, the solution [arXiv:2011.01415] becomes, along a ray in the direction normal to one of the three edges of the initial cloud, a freely propagating onedimensional shock wave of a class proposed by Damski in [Phys. Rev. A 69, 043610 (2004)]. There, for a broad class of initial conditions, the onedimensional hydrodynamic equations can be mapped to the inviscid Burgers' equation, a nonlinear transport equation. More specifically, under the Damski map, the $t=0$ singularity of the original problem becomes, verbatim, the initial condition for the wave catastrophe solution found by Chandrasekhar in 1943 [Ballistic Research Laboratory Report No. 423 (1943)]. At $t=T/8$, our interpretation ceases to exist: at this instance, all three effectively onedimensional shock waves emanating from each of the three sides of the initial triangle collide at the origin, and the 2D1D correspondence between the solution of [arXiv:2011.01415] and the DamskiChandrasekhar shock wave becomes invalid.
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Submission & Refereeing History
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Reports on this Submission
Anonymous Report 2 on 2021423 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2102.12184v5, delivered 20210423, doi: 10.21468/SciPost.Report.2832
Strengths
1  Nice problem, clearly exposed.
2  An elegant combination of interesting and nontrivial concepts applied to a beautiful experiment.
Weaknesses
none
Report
This is a theoretical work based on a recent beautiful experiment of Dalibard's group in Paris (ref.[1]) and on a previous theoretical article by Shi, Gao and Zhai (ref.[5]). The experiment proved the existence of triangularshaped breathers in 2D harmonically trapped Bose condensates, while the authors of ref.[5] found a quite surprising map between the hydrodynamic equations governing these 2D breathers and those describing a highdensity zerotemperature triangularshaped cloud of fermions. Such a mapping, however, has a problem at the initial time, since it implies an infinite force at the cloud edge. In the present work, a solution is found to this problem involving shock waves and making a connection with an old prediction of Chandrasekhar. The three papers together provide a clean example of clever application of general concepts of symmetry, scale invariance and nonlinearity in the collective dynamics of manybody quantum systems, accessible to experimental investigation.
The quality of the paper is high, both for the relevance of the results and the clarity of presentation.
I recommend publication after very minor changes (see below).
Requested changes
1  in the definition of the velocity v, I suggest to enclose the gradient of Psi and Psi* in parenthesis, in order to avoid misunderstanding, especially in the second term.
2  near the end of section 4, replace "clout" with "cloud"
3  after eq.(9), replace "Reacall" with "Recall"
4  at the beginning of section 5, replace "Pitaevski" with "Pitaevskii"
5  six lines after, correct "separates form..."
6  after eq.(10) the authors say: "it can be shown that such a truncated function remains a solution of the inviscid Burgers’ equation". Is it easy to demonstrate? Should the reader take it for granted?
Anonymous Report 1 on 2021419 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2102.12184v5, delivered 20210419, doi: 10.21468/SciPost.Report.2804
Strengths
1 There is a clever idea behind whole this work supported by physically and technicallysignificant results on a hot research topic.
2 The presentation of these findings is concise, without unnecessary discussions and propaganda. The paper is easy to follow.
3 The plots are adequate, especially the first one, cleverly putting different pieces of the considered setup into one place.
4 The reference to Chandrasekhar’s paper is real nice, I have never seen it cited before in the context of cold atom shock waves.
Weaknesses
1 The manuscript does not discuss the role of the quantum pressure term in the dynamics of the studied system.
Report
The manuscript “Triangular GrossPitaevskii breathers and DamskiChandrasekhar shock waves” analyzes the outcome of the recent, already wellknown, experiment of the Paris group from the perspective of the shock wave phenomenon.
It was found in this fascinating experiment that initially triangular and discshaped spatially flat twodimensional clouds of atoms undergo robust periodic dynamics in a harmonic potential. This work triggers (at the very least) the following two outstanding questions
$\bullet$ why other initial shapes evolve differently,
$\bullet$ how dynamics of twodimensional condensates, starting from a flat density profile bounded by a sharp edge, proceed.
The first of these questions is interestingly analyzed in the very recent, possibly seminal, paper of Shi, Gao & Zhai. The second one is discussed in the manuscript that is under review here.
In this manuscript, the authors very interestingly argue that some aspects of the dynamics of the abovementioned condensate can be successfully explained by invoking the shock wave phenomenon. They have written down an elegant analytical 1+1 dimensional shock wave solution, described the mapping/conditions allowing for its application to the 2+1 dimensional flat triangular cloud of atoms, and explained where and when it exactly reproduces the elegant result of Shi, Gao & Zhai.
This way, in my opinion, they have added crucial new insights to our understanding of both the outcome of the Paris experiment and the content of the Shi, Gao & Zhai result. Indeed, neither the paper of the Paris group nor the one of Shi, Gao & Zhai (the ones cited in the manuscript) invokes the shock wave phenomenon as far as I can see. This makes the manuscript of Olshanii et al. physicallyinteresting. It can be also argued that this work is technically interesting as well, due to the abovementioned solution that is worked out there. It is my opinion that these two aspects make the manuscript under review wellworth publication in a respectable journal such as SciPost.
As far as technical remarks go, I would like to mention that I have been able to check some equations. In particular, I have quickly verified all equations from Sec. 4, finding them to be correct. A few misprints, which I have been able to locate, are listed along with some remarks/suggestions in the "Requested changes" section.
Summarizing, it is my opinion that after small corrections, the manuscript should be published in SciPost. It does present original insights/results on a hot research topic and it is well structured. Moreover, it may be worth to mention that it seems to be reasonable to assume that this work will trigger interesting subsequent studies. One of the reasons for saying so is that the question of why the discussed hydrodynamic solution seems to work so well, despite the fact that it is written without taking into account the quantum pressure term, remains to be resolved.
Requested changes
1 Given the fact that whole Sec. 1 is composed of subsection 1.1, one may consider removal of the subsection title.
2 One may consider providing explicit definition of SW and SGZ (despite the fact that these terms should be understandable per se).
3 One may indicate below eq. (1) that g is greater than zero, which is tacitly assumed in the manuscript.
4 The last equation of Sec. 1 mentions “ground state of an equilateraltriangleshaped corral” while the text above it says that we deal with “a uniformly filled equilateral triangle”. The two statements are not equivalent due to the boundary effects “supported” by eq. (1).
5 Bold zero in the continuity eq. (2) should be replaced by “normal” zero.
6 “Normal” $\nabla$ in the equation for the velocity field, the one between eqs. (2) and (3), should be replaced by bold $\nabla$.
7 Oddlooking “and” inbetween equations on p. 5 and in eq. (14) perhaps could be removed.
8 “Euler equation” appears by the end of Sec. 2, but it is defined in eq. (7) of Sec. 4. Similarly, “force” is discussed on p. 5, but a precise meaning to it is given on p. 7. One may consider explicit introduction of these terms around eq. (2).
9 It seems to me that the factor of “$\omega$” may be lost in eq. (6), the dimension of the expression on its righthand side is wrong.
10 It might be worth to mention that both the continuity equation and the Euler equation, the ones listed as (7), reduce to eq. (8) under the discussed mapping.
11 I am not sure what the sentence “Formally, this solution is the $\delta t\to0$ limit of the solution generated by ...” exactly means. Could this be clarified?
12 It looks like “also at constant velocity”, which can be found on p. 7, should be replaced by something like “also moves at a constant velocity”.
13 Time argument might be added to $\dot{\cal R}$ in the first equation on p. 9.
14 On the righthand side of the equation on p. 10, the one right above the sentence “The identification of System 1 is completed by setting”, a square root seems to be missing.
15 It seems to me that in eq. (27), the one where the righthand side is just $V_\mu$, one should have $V_\mu^2$ instead.
Author: Vanja Dunjko on 20210505 [id 1408]
(in reply to Report 1 on 20210419)We thank the Referee for the positive review and encouragement. We have implemented all the recommendations the Referee has made, and we thank the Referee for those as well.
Author: Vanja Dunjko on 20210505 [id 1409]
(in reply to Report 2 on 20210423)We thank the Referee for the positive review and encouragement. We have implemented all the recommendations the Referee has made, and we thank the Referee for those as well.