Scattering for a 3D coupled nonlinear Schr\"odinger system

TL;DR

This paper proves that solutions to a 3D coupled nonlinear Schrödinger system with initial data below certain thresholds are global and scatter, extending techniques from single-equation cases to coupled systems.

Contribution

It establishes scattering results for the 3D coupled NLS system under conditions involving mass and energy thresholds, generalizing previous single-equation results.

Findings

Solutions are global and scatter under specified conditions.

Thresholds involve mass and energy relative to ground states.

Method adapts techniques from single NLS to coupled systems.

Abstract

We consider the three-dimensional cubic nonlinear Schr\"odinger system \begin{equation*} \begin{cases} i\partial_tu+\Delta u+(|u|^2+\beta |v|^2)u=0,\\ i\partial_tv+\Delta v+(|v|^2+\beta |u|^2)v=0. \end{cases} \end{equation*} Let $(P,Q)$ be any ground state solution of the above Schr\"odinger system. We show that for any initial data $(u_0,v_0)$ in $H^1(\mathbb{R}^3)\times H^1(\mathbb{R}^3)$ satisfying $M(u_0,v_0)A(u_0,v_0)<M(P,Q)A(P,Q)$ and $M(u_0,v_0)E(u_0,v_0)<M(P,Q)E(P,Q)$, where $M(u,v)$ and $E(u,v)$ are the mass and energy (invariant quantities) associated to the system, the corresponding solution is global in $H^1(\mathbb{R}^3)\times H^1(\mathbb{R}^3)$ and scatters. Our approach is in the same spirit of Duyckaerts-Holmer-Roudenko, where the authors considered the 3D cubic nonlinear Schr\"odinger equation.