2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/224411We consider a nonlinear Schroedinger equation with a finite bands periodic potential in R . We assume the existence of an orbitally stable family of ground states. We prove that under appropriate hypotheses the ground states are asymptotically stable.The present version of the paper, has more details than the one to be published on the Tansactions AMSAnalysis of PDEstext