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

Scattering of radial $H^1$ solutions to the 3D focusing cubic nonlinear Schrö\-din\-ger equation below a mass-energy threshold $M[u]E[u] <M[Q]E[Q]$ and satisfying an initial mass-gradient bound $\Vert u_0 \Vert_{L^2} \Vert \nabla u_0 \Vert_{L^2} <\Vert Q \Vert_{L^2} \Vert \nabla Q \Vert_{L^2}$, where $Q$ is the ground state, was established in Holmer-Roudenko \cite{HR2}. In this note, we extend the result in \cite{HR2} to non-radial $H^1$ data. For this, we prove a non-radial profile decomposition involving a spatial translation parameter. Then, in the spirit of Kenig-Merle \cite{KM06b}, 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.

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Published 1 January 2008