M. Olshanii, D. Deshommes, J. Torrents, M. Gonchenko, V. Dunjko, G. E. Astrakharchik
SciPost Phys. 10, 114 (2021) ·
published 26 May 2021

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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 mapusing the scale invariance of the problemthe 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, 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 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.