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

M. Olshanii, D. Deshommes, J. Torrents, M. Gonchenko, V. Dunjko, G. E. Astrakharchik

SciPost Phys. 10, 114 (2021) · published 26 May 2021

- doi: 10.21468/SciPostPhys.10.5.114
- Submissions/Reports

### 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.

### Cited by 2

### Authors / Affiliations: mappings to Contributors and Organizations

See all Organizations.-
^{1}Maxim Olshanii, -
^{1}Dumesle Deshommes, -
^{2}Jordi Torrents, -
^{3}Marina Gonchenko, -
^{1}Vanja Dunjko, -
^{3}G. E. Astrakharchik

^{1}University of Massachusetts Boston / University of Massachusetts Boston^{2}Universitat de Barcelona / University of Barcelona [UB]^{3}Universitat Politècnica de Catalunya [UPC]

- Generalitat de Catalunya / Government of Catalonia
- Ministerio de Ciencia e Innovación
- Ministerio de Economía y Competitividad (MINECO) (through Organization: Ministerio de Economía, Industria y Competitividad / Ministry of Economy, Industry and Competitiveness [MINECO])
- National Science Foundation [NSF]
- United States - Israel Binational Science Foundation (through Organization: United States-Israel Binational Science Foundation [BSF])