  In this paper, we study the global dynamics of a class of nonlinear
Schr\"odinger equations using perturbative and non-perturbative methods. We
prove the semi-global existence of solutions for initial conditions close to
constant. That is, solutions will exist for all positive time or all negative
time. The existence of an open set of initial data which limits to zero in both
forward and backward time is also demonstrated. This result in turn forces the
non-existence of any real-analytic conserved quantities. For the quadratic
case, we prove the existence of two (infinite) families of nontrivial unstable
equilibria and prove the existence of heteroclinic orbits limiting to the
nontrivial equilibria in backward time and to zero in forward time. By a time
reversal argument, we also obtain heteroclinic orbits limiting to the
nontrivial equilibria in forward time and to zero in backward time. The proofs
for the quadratic equation are computer-assisted and rely on three separate
ingredients: an enclosure of a local unstable manifold at the equilibria, a
rigorous integration of the flow (starting from the unstable manifold) and a
proof that the solution enters a validated stable set (hence showing
convergence to zero).
