Asymptotic stability in the energy space for dark solitons of the   Gross-Pitaevskii equation
  We pursue our work on the dynamical stability of dark solitons for the one-dimensional Gross-Pitaevskii equation. In this paper, we prove their asymptotic stability under small perturbations in the energy space. In particular, our results do not require smallness in some weighted spaces or a priori spectral assumptions. Our strategy is reminiscent of the one used by Martel and Merle in various works regarding generalized Korteweg-de Vries equations. The important feature of our contribution is related to the fact that while Korteweg-de Vries equations possess unidirectional dispersion, Schr\"odinger equations do not. 
Fabrice B\'ethuel (LJLL), Philippe Gravejat (CMLS-EcolePolytechnique),   Didier Smets (LJLL)
math.AP