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A theory of time dependent nonlinear dispersive equations of the Schroedinger/ Gross-Pitaevskii and Hartree type is developed. The short, intermediate andlarge time behavior is found, by deriving nonlinear Master equations (NLME),governing the evolution of the mode powers, and by a novel multi-time scaleanalysis of these equations. The scattering theory is developed and coherentresonance phenomena and associated lifetimes are derived. Applications includeBEC large time dynamics and nonlinear optical systems. The theory reveals anonlinear transition phenomenon, ``selection of the ground state'', and NLMEpredicts the decay of excited state, with half its energy transferred to theground state and half to radiation modes. Our results predict the recentexperimental observations of Mandelik et. al. in nonlinear optical waveguides

Theory of Nonlinear Dispersive Waves and Selection of the Ground State