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

Long time dynamics near the symmetry breaking bifurcation for nonlinear Schr\"odinger/Gross-Pitaevskii Equations