  We consider a class nonlinear Schr\"odinger / Gross-Pitaevskii equations
(NLS/GP) with a focusing (attractive) nonlinear potential and symmetric double
well linear potential. NLS/GP plays a central role in the modeling of nonlinear
optical and mean-field quantum many-body phenomena. It is known that there is a
critical $L^2$ norm (optical power / particle number) at which there is a
symmetry breaking bifurcation of the ground state. We study the rich dynamical
behavior near the symmetry breaking point. The source of this behavior in the
full Hamiltonian PDE is related to the dynamics of a finite-dimensional
Hamiltonian reduction. We derive this reduction, analyze a part of its phase
space and prove a {\it shadowing theorem} on the persistence of solutions, with
oscillating mass-transport between wells, on very long, but finite, time scales
within the full NLS/GP. The infinite time dynamics for NLS/GP are expected to
depart, from the finite dimensional reduction, due to resonant coupling of
discrete and continuum / radiation modes.
