Abstract
Scattering of radial $H^1$ solutions to the 3D focusing cubic nonlinear Schr\o\-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.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.