Scattering for the nonradial 3D cubic nonlinear Schroedinger equation
Abstract
Scattering of radial $H^1$ solutions to the 3D focusing cubic nonlinear Schrödinger equation below a massenergy threshold $M[u]E[u] < M[Q]E[Q]$ and satisfying an initial massgradient bound $\u_0\_{L^2} \\nabla u_0 \_{L^2} < \Q\_{L^2} \\nabla Q\_{L^2}$, where $Q$ is the ground state, was established in HolmerRoudenko (2007). In this note, we extend the result in HolmerRoudenko (2007) to nonradial $H^1$ data. For this, we prove a nonradial profile decomposition involving a spatial translation parameter. Then, in the spirit of KenigMerle (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.
 Publication:

arXiv eprints
 Pub Date:
 October 2007
 arXiv:
 arXiv:0710.3630
 Bibcode:
 2007arXiv0710.3630D
 Keywords:

 Mathematics  Analysis of PDEs