Scattering for a 3D coupled nonlinear Schrödinger system

We consider the three-dimensional cubic nonlinear Schrödinger system \begin{equation*} \begin{cases} i\partial_tu+Δu+(|u|^2+β|v|^2)u=0,\\ i\partial_tv+Δv+(|v|^2+β|u|^2)v=0. \end{cases} \end{equation*} Let $(P,Q)$ be any ground state solution of the above Schrödinger 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 Schrödinger equation.

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