Vortices in rotating Bose Einstein condensates

One of the key issues related to superfluidity is the existence of quantized vortices. We present very recent experiments on Bose-Einstein condensates exhibiting vortices, which consist in rotating the trap holding the atoms. We investigate the behavior of the wave function which minimizes the Gross Pitaevskii energy. This energy takes into account the special shape of the trapping potential. In a regime with a small parameter, we give a simplified expression of the energy which only depends on the number and shape of vortex lines. This allows us to study in detail the structure of the lines which have either a $U$ or $S$ shape and compare with experiments. We also present results where the type of trapping potential can be at the origin of multiply quantized vortices. Finally, we will mention another type of experiments where the condensate flows around an obstacle and we are interested in existence of vortex free solutions.