  We place a molecular Bose-Einstein condensate in a 1D shaken lattice with a
Floquet-engineered dispersion, and observe the dynamics in both position and
momentum space. At the initial condition of zero momentum, our engineered
dispersion is inverted, and therefore unstable. We observe that the condensate
is destabilized by the lattice shaking as expected, but rather than decaying
incoherently or producing jets, as in other unstable condensates, under our
conditions the condensate bifurcates into two portions in momentum space, with
each portion subsequently following semi-classical trajectories that suffer
minimal spreading in momentum space as they evolve. We can model the evolution
with a Gross-Pitaevskii equation, which suggests the initial bifurcation is
facilitate by a nearly linear "inverted V"-shaped dispersion at the zone
center, while the lack of spreading in momentum space is facilitated by
interactions, as in a soliton. We propose that this relatively clean
bifurcation in momentum space has applications for counter-diabatic preparation
of exotic ground states in many-body quantum simulation schemes.
