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

TL;DR

This paper extends scattering results for the 3D focusing cubic nonlinear Schrödinger equation from radial to non-radial initial data by developing a non-radial profile decomposition and employing a virial-based argument.

Contribution

It introduces a non-radial profile decomposition with spatial translation and adapts the Kenig-Merle approach to prove scattering for non-radial data.

Findings

Established scattering for non-radial $H^1$ solutions below the mass-energy threshold.

Developed a non-radial profile decomposition involving spatial translation.

Controlled the divergence rate of translation via momentum conservation and virial arguments.

Abstract

Scattering of radial $H^1$ solutions to the 3D focusing cubic nonlinear Schr\"odinger equation below a mass-energy threshold $M[u]E[u] < M[Q]E[Q]$ and satisfying an initial mass-gradient 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 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.