Quantum ripples over a semi-classical shock

The evolution of an initially smooth spatial inhomogeneity in the density of a one-dimensional Fermi gas is well described by classical mechanics. The classical evolution leads to the formation of a shock wave: the density develops kinks in its coordinate dependence. We show that quantum corrections to the shock wave produce density ripples which run off the kinks. Despite their quantum origin, the amplitude and period of the ripples are expressed only in terms of classical objects derived from a smooth density profile.