09/09/2016, 12:00 — 12:40 — Amphitheatre Pa2, Mathematics Building
Ademir Pastor, Universidade Estadual de Campinas

Scattering for a 3D Coupled Nonlinear Schrödinger System

We consider the three-dimensional cubic nonlinear Schrödinger system \[ \begin{cases} i \partial_t u + \triangle u + (|u|^2+\beta |v|^2)u = 0 , \\ i \partial_t v + \triangle v + (|v|^2+\beta |u|^2)v = 0 . \end{cases} \] 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. Joint work with L.G. Farah (UFMG).