On traveling waves for some Gross-Pitaevskii equations

In this talk, we will discuss some properties of traveling waves solutions for some variants of the classical Gross-Pitaevskii equation in the whole space, in order to include new physical models in Bose-Einstein condensates and nonlinear optics. We are interested in the existence of finite energy localized traveling waves solutions with nonvanishing conditions at infinity, i.e. dark solitons. After a review of the state of the art in the classical case, we will show some results for a family of Gross-Pitaevskii equations with nonlocal interactions in the potential energy, obtained by variational techniques. Then, we will discuss the existence and behavior of the dark solitons for the Gross-Pitaevskii equation is a strip, according to its width.

This is joint work with Philippe Gravejat, Salvador Lopez-Martinez, and Didier Smets.