  We use the ab initio Bethe Ansatz dynamics to predict the dissociation of
one-dimensional cold-atom breathers that are created by a quench from a
fundamental soliton. We find that the dissociation is a robust quantum
many-body effect, while in the mean-field (MF) limit the dissociation is
forbidden by the integrability of the underlying nonlinear Schr\"{o}dinger
equation. The analysis demonstrates the possibility to observe quantum
many-body effects without leaving the MF range of experimental parameters. We
find that the dissociation time is of the order of a few seconds for a typical
atomic-soliton setting.
