# Triangular Gross-Pitaevskii breathers and Damski-Chandrasekhar shock waves

### Submission summary

 As Contributors: Vanja Dunjko · Maxim Olshanii Preprint link: scipost_202105_00006v1 Date accepted: 2021-05-17 Date submitted: 2021-05-05 18:34 Submitted by: Dunjko, Vanja Submitted to: SciPost Physics Academic field: Physics Specialties: Atomic, Molecular and Optical Physics - Theory Approach: Theoretical

### Abstract

The recently proposed map [arXiv:2011.01415] between the hydrodynamic equations governing the two-dimensional triangular cold-bosonic breathers [Phys. Rev. X 9, 021035 (2019)] and the high-density zero-temperature triangular free-fermionic 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 Fermi-Bose 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 one-dimensional shock wave of a class proposed by Damski in [Phys.~Rev.~A 69, 043610 (2004)]. There, for a broad class of initial conditions, the one-dimensional hydrodynamic equations can be mapped to the inviscid Burgers' equation, which is equivalent to 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 one-dimensional shock waves emanating from each of the three sides of the initial triangle collide at the origin, and the 2D-1D correspondence between the solution of [arXiv:2011.01415] and the Damski-Chandrasekhar shock wave becomes invalid.

Published as SciPost Phys. 10, 114 (2021)

We thank both referees for the positive reviews and encouragement, and for pointing out the places in the manuscript where clarifications or rephrasings were needed and where typographical errors had to be corrected. We have implemented all of their recommendations in our resubmission.

### List of changes

In the abstract, we reformulated the relationship between the inviscid Burgers' equation and the nonlinear transport equation, replacing "the inviscid Burgers' equation, a nonlinear transport equation" with "the inviscid Burgers' equation, which is equivalent to a nonlinear transport equation."

We demoted the former sections 2 and 3 to subsections of the introduction. We also renamed the first subsection of the introduction.

In the caption of Figure 1, we added the explanation that "SW" means "shock wave".

In the name of the penultimate section, we replaced 'SGZ' with 'Shi-Gao-Zhai'.

Below Eq. (1), we indicated that g is positive.

In the description of the ground state following Eq. (1), we removed the comment, which is incorrect at the GPE level, that the ground state fills the corral with a uniform density.

In the 'equation' for the ground-state wavefunction that follows, we added the definite article to the description on the right-hand side.

We also moved the references that used to appear right before this equation so that now they are after it.

In Eq. (2), we identified the first equation as the continuity equation, and the second as the Euler equation.

In Eq. (2), the right-hand side of the continuity equation now has a zero that is not in boldface (i.e. it is a scalar).

In Eq. (2), in the last term on the right-hand side of the Euler equation, the variable r is now in boldface (i.e. it is a vector).

In the equation for the velocity field that follows Eq. (2), the nabla (i.e. del) symbol is now in boldface.

Moreover, the gradients in that equation are now enclosed in brackets, while the outermost delimiters are now large parentheses rather than brackets.

In Eq. (3), the right-hand side of the velocity field is now a zero vector rather than just a scalar zero.

We removed the word 'and' following the penultimate equation in the list of equations on page 5. Similarly, we removed the word 'and' in the former Eq. (14), now Eq. (15).

In the last paragraph of Sec. 1.2, we now explicitly say that the interaction force term in the Euler equation is -nabla (g n).

In the same paragraph, we replaced 'At t=0 and at t=T/4, the solution possesses density jump, at the edge of the clout' with 'At t=0 and t=T/4, the solution has, at the edge of the cloud, a discontinuous jump in the density'.

Also, previously we said that this term is on the left-hand side of the Euler equation; we corrected that to say the right-hand side.

In the following paragraph, we now explicitly say that the force of the external trap is -m omega^2 r.

We added the missing factor of omega on the right-hand side of Eq. (6).

In the text leading up to Eq. (8), we now emphasize that the ansatz under discussion turns both the continuity equation and the Euler equation into the inviscid Burgers’ equation.

In Eq. (10) (formerly (9)), we added the subscript 'Chandrasekhar' to u(z,t).

We have added an explanation of what we mean by the delta t -> 0 limit in the discussion following Eq. (10) (formerly (9)).

In point 3. on p. 7, we replaced 'also at constant velocity:' by 'also moves at the constant velocity'.

We added the time argument to all the appearances of the hyperradius R in the equation following Eq. (18) (formerly (17)), and in Eq. (19) (formerly (18)).

On p. 10, in the equation right above the phrase 'The identification of System 1 is completed by setting', the right-hand side is now the square root of what it was previously.

In the middle one of the three equations in Eq. (28) (formerly (27), we now have the square of V_mu rather than just V_mu.

In the paragraph following Eq. (8), we corrected the typo 'Reacall' (it's 'Recall').

At the beginning of Sec. 3 (formerly Sec. 5), we corrected the typo 'Pitaevski' (it's 'Pitaevskii').

In the same paragraph, we corrected the typo 'separates form' (it's 'separates from').

In the text following Eq. (11) (formerly (10)), we now explain why the truncated function that we mention remains a solution of the inviscid Burgers’ equation.

In Eq. (26) (formerly (25)), we removed the explicit time argument of the hyperradius R_2

In Eq. (27) (formerly (26)), we added the explicit argument t to the function t_2.

In the final section, we replaced 'the inviscid Burgers’ equation (which is a nonlinear transport equation)' with 'the inviscid Burgers’ equation (also commonly referred to as the nonlinear transport equation)'.

### Submission & Refereeing History

Resubmission scipost_202105_00006v1 on 5 May 2021
Submission 2102.12184v5 on 29 March 2021

## Reports on this Submission

### Report

My remarks from the previous report have been properly addressed.

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