Hamiltonian Hopf bifurcations and dynamics of NLS/GP standing-wave modes

We examine the dynamics of solutions to nonlinear Schrödinger/ GrossPitaevskii equations that arise due to semisimple indefinite Hamiltonian Hopf bifurcations-the collision of pairs of eigenvalues on the imaginary axis. We construct localized potentials for this model which lead to such bifurcations in a predictable manner. We perform a formal reduction from the partial differential equations to a small system of ordinary differential equations. We analyze the equations to derive conditions for this bifurcation and use averaging to explain certain features of the dynamics that we observe numerically. A series of careful numerical experiments are used to demonstrate the phenomenon and the relations between the full system and the derived approximations.