Alberto Farina 
Stationary and Time Dependent Gross-Pitaevskii Equations 

This volume is based on a thematic program on the Gross-Pitaevskii equation, which was held at the Wolfgang Pauli Institute in Vienna in 2006. The program consisted of two workshops and a one-week Summer School. The Gross-Pitaevskii equation, an example of a defocusing nonlinear Schroedinger equation, is a model for phenomena such as the Bose-Einstein condensation of ultra cold atomic gases, the superfluidity of Helium II, or the "dark solitons" of Nonlinear Optics. Many interesting and difficult mathematical questions associated with the Gross-Pitaevskii equation, linked for instance to the nontrivial boundary conditions at infinity, arise naturally from its modeling aspects. The articles in this volume review some of the recent developments in the theory of the Gross-Pitaevskii equation. In particular the following aspects are considered: modeling of superfluidity and Bose-Einstein condensation, the Cauchy problem, the semi-classical limit, scattering theory, existence and properties of coherent traveling structures, and numerical simulations.