Orbital stability of a traveling wave of the Gross-Pitaevskii equation
Fr茅d茅ric Valet, Assistant Professor @ Laboratoire de Math茅matiques dOrsay, University of Paris-Saclay
Abstract:
The two-dimensional Gross-Pitaevskii equation is a non-linear model for the distribution of Bose-Einstein condensates. Despite the similarity with the non-linear Schr篓odinger equation (NLS), the non-vanishing condition at infinity induces different inherent structures from the ones of NLS. Unlike the one-dimensional case, it is also not kwown if this non-linear dispersive equation is integrable and the momentum is defined only formally. Furthermore, a balance between the dispersion and the non-linearity provide traveling waves. The only traveling waves with small velocities behave as a vortex-antivortex pair. We approach in the talk the question of orbital stability of those traveling waves. We introduce a new proof by tackling the problem of the definition of the momentum and by detailing how the non-linearity strongly influences the 鈥漲uadratic鈥 form at infinity. The talk is based on a collaboration with Philippe Gravejat and Eliot Pacherie (CY Cergy Paris Universit茅).