  Using Time-Dependent Density Matrix Renormalization Group (TDMRG) we study
the collision of one-dimensional atomic clouds confined in a harmonic trap and
evolving with the Lieb-Liniger Hamiltonian. It is observed that the motion is
essentially periodic with the clouds bouncing elastically, at least on the time
scale of the first few oscillations that can be resolved with high accuracy.
This is in agreement with the results of the "quantum Newton cradle" experiment
of Kinoshita et al. [Nature 440, 900 (2006)]. We compare the results for the
density profile against a hydrodynamic description, or generalized nonlinear
Schr\"odinger equation, with the pressure term taken from the Bethe Ansatz
solution of the Lieb-Liniger model. We find that hydrodynamics can describe the
breathing mode of a harmonically trapped cloud for arbitrary long times while
it breaks down almost immediately for the collision of two clouds due to the
formation of shock waves (gradient catastrophe). In the case of the clouds'
collision TDMRG alone allows to extract the oscillation period which is found
to be measurably different from the breathing mode period. Concomitantly with
the shock waves formation we observe a local energy distribution typical of
population inversion, i.e., an effective negative temperature. Our results are
an important step towards understanding the hydrodynamics of quantum many-body
systems out of equilibrium and the role of integrability in their dynamics.
