Defocusing nonlinear Schrödinger equation: confinement, stability and asymptotic stability

This paper is devoted to the asymptotic properties of the Nonlinear Schro ̈dinger equation in the defocusing case. We recover dispersion rates by the mean of time-dependent rescalings in the power law case and prove a new result in the case of the logarithmic Nonlinear Schrödinger equation. The rescaled equation is then used to obtain an asymptotic result of nonlinear stability, which is the main result of this paper.