Bose–Einstein condensation was predicted by S.N. Bose and Albert Einstein in 1925: for a gas of non-interacting particles, below a certain temperature there is a phase transition to a localized (condensed) state of lowest energy. This phenomenon was realized experimentally in 1995 in alkali gases by E. Cornell and C. Wieman in Boulder as well as by W. Ketterle at MIT, who all shared the Nobel Prize in 2001. Since that time, the attention of many mathematicians has turned to the analysis of the mean-field model of this phenomenon, which is known as the Gross–Pitaevskii equation or the nonlinear Schrödinger equation with an external potential.

Various trapping mechanisms of Bose–Einstein condensation were realized experimentally, including a parabolic magnetic confinement and a periodic optical lattice. This book is about the Gross–Pitaevskii equation with a periodic potential, in particular about the localized modes supported by the periodic potential. The book is written for young researchers in applied mathematics and so it has the main emphasis on the mathematical properties of the Gross–Pitaevskii equation. It can nevertheless serve as a reference for theoretical physicists interested in the phenomenon of localization in periodic potentials.

Compared to recent work by Lieb et al on the justification of the Gross–Pitaevskii equation as the mean-field approximation of the linear N-body Schrödinger equation [131], this book takes the Gross–Pitaevskii equation as the starting point of analysis.