results, in collaboration with André de Laire, on the existence and properties of "dark solitons", which have been observed for instance in Bose-Einstein condensates. In our context, dark solitons are finite-energy traveling wave solutions that do not vanish at infinity. The proof of the existence is performed in two stages: first, we prove existence for almost every subsonic wave speed via a Mountain-Pass argument, and second, we fill the whole range of speeds for a family of potentials that provide suitable a priori estimates. We show also 
some properties of the solitons such as exponential decay or analyticity. We will finish by discussing a few open problems.