  We study the long-time behavior of solutions to the nonlinear Schr\"odinger /
Gross-Pitaevskii equation (NLS/GP) with a symmetric double-well potential,
continuing work of the 2nd and 3rd authors. NLS/GP governs nearly-monochromatic
guided optical beams in weakly coupled waveguides with both linear and
nonlinear (Kerr) refractive indices and zero absorption. The optical power
($L^2$ norm) is conserved with propagation distance. At low optical power, the
beam energy executes beating oscillations between the two waveguides. There is
an optical power threshold above which the set of guided mode solutions splits
into two families of solutions. One type of solution corresponds to an optical
beam which is concentrated in either waveguide, but not both. Solutions in the
second family undergo tunneling oscillations between the two waveguides. NLS/GP
can also model the behavior of Bose-Einstein condensates. A finite dimensional
reduction (system of ODEs) well-approximates the PDE dynamics on long time
scales. In particular, we derive this reduction, find a class of exact
solutions and prove the very long-time shadowing of these solutions by applying
the approach of the 2nd and 3rd authors.
