Localized States and Dynamics in the Nonlinear Schroedinger / Gross-Pitaevskii Equation

This article is a review of results on the nonlinear Schroedinger / Gross-Pitaevskii equation (NLS / GP). Nonlinear bound states and aspects of their stability theory are discussed from variational and bifurcation perspectives. Nonlinear bound states, in the particular cases where the potential is a single-well, double-well potential and periodic potential, are discussed in detail. We then discuss particle-like dynamics of solitary wave solutions interacting with a potential over a large, but finite, time interval. Finally we turn to the very long time behavior of solutions. We focus on the important resonant radiation damping mechanism that drives the relaxation of the system to a nonlinear ground state and underlies the phenomena of {\it Ground State Selection} and {\it Energy Equipartition}. We also analyze linear and nonlinear "toy minimal models", which illustrate these mechanisms. Regarding the overall style of this article, we seek to emphasize the key ideas and therefore do not present all technical details, leaving that to the references.