The recently proposed map [5] between the hydrodynamic equations
governing the two-dimensional 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<0t<0.
A similar singularity appears at
t = T/4t=T/4,
where T
is the period of the harmonic trap, with the Fermi-Bose map becoming
invalid at
t > T/4t>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/8t=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 [5] and the Damski-Chandrasekhar
shock wave becomes invalid.
Triangular Gross-Pitaevskii breathers and Damski-Chandrasekhar shock waves
