André de Laire (Université de Lille, Laboratoire Paul Painlevé): Minimizing travelling waves for the Gross-Pitaevskii equation on the 2D strip

We investigate the two-dimensional defocusing nonlinear cubic Schrödinger (Gross–Pitaevskii) equation with nonzero conditions at infinity, on a strip, i.e. on an infinite channel of finite transverse width. We establish the existence of traveling waves that minimize the Ginzburg–Landau energy at fixed momentum. We establish a sharp bifurcation from planar to multidimensional behavior. Precisely, we show that there exists a threshold value for the transverse width below which minimizers are the one-dimensional dark solitons (planar solitons), and above which they are genuinely two-dimensional dark solitons.
This is a joint work with Didier Smets and Philippe Gravejat.