A Bose condensate subject to a periodic modulation of the two-body interactions was recently observed to emit matter-wave jets resembling fireworks [Nature 551, 356(2017)]. In this paper, combining experiment with numerical simulation, we demonstrate that these Bose fireworks represent a late stage in a complex time evolution of the driven condensate. We identify a density wave stage which precedes jet emission and results from interference of matterwaves. The density waves self-organize and self-amplify without the breaking of long range translational symmetry. Importantly, this density wave structure deterministically establishes the template for the subsequent patterns of the emitted jets. Our simulations, in good agreement with experiment, also address the apparent asymmetry in the jet pattern and show it is fully consistent with momentum conservation.
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