Scattering for the non-radial 3D cubic nonlinear schrödinger equation

Abstract

Scattering of radial H1 solutions to the 3D focusing cubic nonlinear Schrö dinger equation below a mass-energy threshold M[u]E[u] < M[Q]E[Q] and satisfying an initial mass-gradient bound ∥u0∥ L2 ∥ ∇u0 ∥ L2 < ∥Q∥ L2 ∥∇Q∥ L2, where Q is the ground state, was established in Holmer-Roudenko [7]. In this note, we extend the result in [7] to non-radial H1 data. For this, we prove a non-radial profile decomposition involving a spatial translation parameter. Then, in the spirit of Kenig-Merle [10], we control via momentum conservation the rate of divergence of the spatial translation parameter and by a convexity argument based on a local virial identity deduce scattering. An application to the defocusing case is also mentioned. © International Press 2008.