  Long time dynamics of the smoothed step initial value problem or dispersive
Riemann problem for the Benjamin-Bona-Mahony (BBM) equation $u_t + uu_x =
u_{xxt}$ are studied using asymptotic methods and numerical simulations. The
catalog of solutions of the dispersive Riemann problem for the BBM equation is
much richer than for the related, integrable, Korteweg-de Vries equation $u_t +
uu_x + u_{xxx} =0.$ The transition width of the initial smoothed step is found
to significantly impact the dynamics. Narrow width gives rise to rarefaction
and dispersive shock wave (DSW) solutions that are accompanied by the
generation of two-phase linear wavetrains, solitary wave shedding, and
expansion shocks. Both narrow and broad initial widths give rise to two-phase
nonlinear wavetrains or DSW implosion and a new kind of dispersive Lax shock
for symmetric data. The dispersive Lax shock is described by an approximate
self-similar solution of the BBM equation whose limit as $t \to \infty$ is a
stationary, discontinuous weak solution. By introducing a slight asymmetry in
the data for the dispersive Lax shock, the generation of an incoherent solitary
wavetrain is observed. Further asymmetry leads to the DSW implosion regime that
is effectively described by a pair of coupled nonlinear Schr\"{o}dinger
equations. The complex interplay between nonlocality, nonlinearity and
dispersion in the BBM equation underlies the rich variety of nonclassical
dispersive hydrodynamic solutions to the dispersive Riemann problem.
