The Nonlinear Schrodinger equation, is an example of a dispersive wave equation which has many different asymptotic states depending on the initial data. Such time dependent equations play a central role in many of the latestscientific advances,such as Bose-Einstein condensates and optical devices .I will discuss the solutions of such equations,including the large time behavior:first rigorous proof of the phenomena of ground state selection, asymptotic instability of the excited states and more.These results are obtained by deriving a novel Nonlinear Master equation and multitime scale analysis of its properties.The talk will be general.