Dark Solitons in Bose-Einstein Condensates: Theory

Abstract

The history of dark solitons starts with the pioneering paper by Tsuzuki [1], where exact soliton solutions of the Gross–Pitaevskii equation (GPE) [2], also known as the nonlinear Schrodinger (NLS) equation, were obtained and their connection with Bogoliubov’s phonons [3] was revealed. The seminal paper by Zakharov and Shabat [4], devoted to Fraunhofer diffraction in a defocusing Kerr medium, was the next fundamental step in understanding the nature of dark solitons. The NLS equation was solved by means of the inverse scattering transform (see also [5, 6]), allowing one to obtain multisoliton solutions and, in particular, to recover Tsuzuki’s soliton. The work has opened new ways for systematic studies of dynamics of dark solitons in the presence of perturbations and of their generation. The perturbation theory for dark solitons was first proposed in [7] and later on in [8] in a different mathematical statement, while investigations of practical aspects of dark soliton generation were initiated by the works [9]. Earlier studies of dark solitons were focused on their optical applications and were stimulated by theoretical prediction [10], where the term “dark” was introduced, and experimental observation [11] of dark solitons in nonlinear optical fibers with normal dispersion. First experimental generation of dark solitons [12] in a Bose–Einstein condensate (BEC) of Rb atoms has stimulated new theoretical studies. Application of the theory to experiments already carried out and prediction of new effects in BECs, include consideration of such factors as multidimensionality of the system, trapping potentials, finite life-times of condensates, specific conditions of soliton generation, interaction of condensate with the thermal cloud, etc. In the present chapter we address only some of these issues, restricting ourselves to quasi-one-dimensional (1D) single-component atomic BECs in elongated traps at zero temperature. For detailed considerations of the topics mentioned above we refer to other chapters of this book, and in particular, Parts V, IX, and XI.