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

Abstract

Abstract. Scattering of radial H1 solutions to the 3D focusing cubic nonlinear Schrödinger 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 [8]. In this note, we extend the result in [8] 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. 1.