Nonlinear waves in Bose–Einstein condensates: physical relevance and mathematical techniques

TLDR

Recent experimental setups in Bose–Einstein condensates still require advanced mathematical and computational tools. This review introduces key physical concepts and mathematical techniques for nonlinear waves in Bose–Einstein condensates and discusses emerging experimental challenges. The authors survey prototypical BEC models across dimensions and potentials, analyze their wave solutions, and compile mathematical methods for studying existence, stability, and dynamics.

Abstract

The aim of the present review is to introduce the reader to some of the physical notions and of the mathematical methods that are relevant to the study of nonlinear waves in Bose-Einstein Condensates (BECs). Upon introducing the general framework, we discuss the prototypical models that are relevant to this setting for different dimensions and different potentials confining the atoms. We analyze some of the model properties and explore their typical wave solutions (plane wave solutions, bright, dark, gap solitons, as well as vortices). We then offer a collection of mathematical methods that can be used to understand the existence, stability and dynamics of nonlinear waves in such BECs, either directly or starting from different types of limits (e.g., the linear or the nonlinear limit, or the discrete limit of the corresponding equation). Finally, we consider some special topics involving more recent developments, and experimental setups in which there is still considerable need for developing mathematical as well as computational tools.