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In this short topical review, we revisit a number of works on the pattern-forming dynamical instabilities of Bose­Einstein condensates in one- and two-dimensional settings. In particular, we illustrate the trapping conditions that allow the reduction of the three-dimensional, mean field description of the condensates (through the Gross­Pitaevskii equation) to such lower dimensional settings, as well as to lattice settings. We then go on to study the modulational instability in one dimension and the snaking/transverse instability in two dimensions as typical examples of long-wavelength perturbations that can destabilize the condensates and lead to the formation of patterns of coherent struc- tures in them. Trains of solitons in one dimension and vortex arrays in two dimensions are prototypical examples of the resulting nonlinear waveforms, upon which we briefly touch at the end of this review. [ABSTRACT FROM AUTHOR]

In this brief review we summarize a number of recent developments in the study of vortices in Bose–Einstein condensates, a topic of considerable theoretical and experimental interest in the past few years. We examine the generation of vortices by means of phase imprinting, as well as via dynamical instabilities. Their stability is subsequently examined in the presence of purely magnetic trapping, and in the combined presence of magnetic and optical trapping. We then study pairs of vortices and their interactions, illustrating a reduced description in terms of ordinary differential equations for the vortex centers. In the realm of two vortices we also consider the existence of stable dipole clusters for two-component condensates. Last but not least, we discuss mesoscopic patterns formed by vortices, the so-called vortex lattices and analyze some of their intriguing dynamical features. A number of interesting future directions are highlighted. [ABSTRACT FROM AUTHOR]