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

Abstract

Scattering of radial $H^1$ 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 $\Vert u_0{\Vert }_{L^2}\Vert \nabla u_0{\Vert }_{L^2}<\Vert Q{\Vert }_{L^2}\Vert \nabla Q{\Vert }_{L^2}$, where $Q$ is the ground state, was established in Holmer-Roudenko (2007). In this note, we extend the result in Holmer-Roudenko (2007) 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 (2006), 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.

Cited by in corpus (2)

• The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher
• Threshold solutions for the focusing 3d cubic Schroedinger equation