Triangular Gross-Pitaevskii breathers and Damski-Chandrasekhar shock waves

The recently proposed map [5] between the hydrodynamic equations governing the twodimensional triangular cold-bosonic breathers [1] 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 [5] 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 [7]. 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 [9]. 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 2D-1D correspondence between the solution of [5] and the Damski-Chandrasekhar shock wave becomes invalid.