  We simulate numerically the free expansion of a repulsive Bose-Einstein
condensate with an initially Gaussian density profile. We find a self-similar
expansion only for weak inter-atomic repulsion. In contrast, for strong
repulsion we observe the spontaneous formation of a shock wave at the surface
followed by a significant depletion inside the cloud. In the expansion,
contrary to the case of a classical viscous gas, the quantum fluid can generate
radial rarefaction density waves with several minima and maxima. These
intriguing nonlinear effects, never observed yet in free-expansion experiments
with ultra-cold alkali-metal atoms, can be detected with the available setups.
