2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/71285We construct a local Lipschitz graph around a soliton of the cubic focusing NLS in three dimensions on which global solutions exist, and asymptotic stability as well as scattering holds.This is a revised version of a paper that was originally posted in May. In this new version, the spectral assumptions have been changed. Only the generic ones remain. Also, there is a new Theorem 2 that establishes the existence of a codimension one stable manifoldAnalysis of PDEsMathematical Physics37K40, 37K45, 35Q51, 35P25, 35P30text