  The strongly interacting regime for attractive Bose-Einstein condensates
(BECs) tightly confined in an extended cylindrical trap is studied. For
appropriately prepared, non-collapsing BECs, the ensuing dynamics are found to
be governed by the one-dimensional focusing Nonlinear Schr\"odinger equation
(NLS) in the semiclassical (small dispersion) regime. In spite of the
modulational instability of this regime, some mathematically rigorous results
on the strong asymptotics of the semiclassical limiting solutions were obtained
recently. Using these results, "implosion-like" and "explosion-like" events are
predicted whereby an initial hump focuses into a sharp spike which then expands
into rapid oscillations. Seemingly related behavior has been observed in
three-dimensional experiments and models, where a BEC with a sufficient number
of atoms undergoes collapse. The dynamical regimes studied here, however, are
not predicted to undergo collapse. Instead, distinct, ordered structures,
appearing after the "implosion", yield interesting new observables that may be
experimentally accessible.
