  We examine the dynamics of solutions to nonlinear
Schrodinger/Gross-Pitaevskii equations that arise due to Hamiltonian Hopf (HH)
bifurcations--the collision of pairs of eigenvalues on the imaginary axis. To
this end, we use inverse scattering to construct localized potentials for this
model which lead to HH bifurcations in a predictable manner. We perform a
formal reduction from the partial differential equations (PDE) to a small
system of ordinary differential equations (ODE). We show numerically that the
behavior of the PDE is well-approximated by that of the ODE and that both
display Hamiltonian chaos. We analyze the ODE to derive conditions for the HH
bifurcation and use averaging to explain certain features of the dynamics that
we observe numerically.
