Spatiotemporal engineering of matter-wave solitons in Bose–Einstein condensates

Since the realization of Bose–Einstein condensates (BECs) trapped in optical potentials, intensive experimental and theoretical investigations have been carried out for bright and dark matter-wave solitons, coherent structures, modulational instability (MI), and nonlinear excitation of BEC matter waves, making them objects of fundamental interest in the vast realm of nonlinear physics and soft condensed-matter physics. Many of these states have their counterparts in optics, as concerns the nonlinear propagation of localized and extended light modes in the spatial, temporal, and spatiotemporal domains. Ubiquitous models, which are relevant to the description of diverse nonlinear media in one, two, and three dimensions (1D, 2D, and 3D), are provided by the nonlinear Schrödinger (NLS), alias Gross–Pitaevskii (GP), equations. In many settings, nontrivial solitons and coherent structures, which do not exist or are unstable in free space, can be created and/or stabilized by means of various management techniques, which are represented by NLS and GP equations with coefficients in front of linear or nonlinear terms which are functions of time and/or coordinates. Well-known examples are dispersion management in nonlinear fiber optics, and nonlinearity management in 1D, 2D, and 3D BEC. Developing this direction of research in various settings, efficient schemes of the spatiotemporal modulation of coefficients in the NLS/GP equations have been designed to engineer desirable robust nonlinear modes. This direction and related ones are the main topic of the present review. In particular, a broad and important theme is the creation and control of 1D matter-wave solitons in BEC by means of combination of the temporal or spatial modulation of the nonlinearity strength (which may be imposed by means of the Feshbach resonance induced by variable magnetic fields) and a time-dependent trapping potential. An essential ramification of this topic is analytical and numerical analysis of MI of continuous-wave (constant-amplitude) states, and control of the nonlinear development of MI. Another physically important topic is stabilization of 2D solitons against the critical collapse, driven by the cubic self-attraction, with the help of temporarily periodic nonlinearity management, which makes the sign of the nonlinearity periodically flipping. In addition to that, the review also includes some topics that do not directly include spatiotemporal modulation, but address physically important phenomena which demonstrate similar soliton dynamics. These are soliton motion in binary BEC, three-component solitons in spinor BEC, and dynamics of two-component 1D solitons under the action of spin–orbit coupling.