  We study the expansion of repulsively interacting Bose-Einstein condensates
(BECs) in shallow one-dimensional potentials. We show for these systems that
the onset of wave chaos in the Gross-Pitaevskii equation (GPE), i.e. the onset
of exponential separation in Hilbert space of two nearby condensate wave
functions, can be used as indication for the onset of depletion of the BEC and
the occupation of excited modes within a many-body description. Comparison
between the multiconfigurational time-dependent Hartree for bosons (MCTDHB)
method and the GPE reveals a close correspondence between the many-body effect
of depletion and the mean-field effect of wave chaos for a wide range of
single-particle external potentials. In the regime of wave chaos the GPE fails
to account for the fine-scale quantum fluctuations because many-body effects
beyond the validity of the GPE are non-negligible. Surprisingly, despite the
failure of the GPE to account for the depletion, coarse grained expectation
values of the single-particle density such as the overall width of the atomic
cloud agree very well with the many-body simulations. The time dependent
depletion of the condensate could be investigated experimentally, e.g., via
decay of coherence of the expanding atom cloud.
