Scattering for a mass critical NLS system below the ground state with and without mass-resonance condition

Abstract

We consider a mass-critical system of nonlinear Schrödinger equations \begin{document}$ \left\{ \begin{split} i\partial_t u + \;\; \Delta u & = \bar{u}v,\\ i\partial_t v +\kappa \Delta v & = u^2, \end{split} \right. \qquad (t,x)\in \mathbb{R}\times \mathbb{R}^4, $\end{document} where $ (u,v) $ is a $ \mathbb{C}^2 $-valued unknown function and $ \kappa >0 $ is a constant. If $ \kappa = 1/2 $, we say the equation satisfies mass-resonance condition. We are interested in the scattering problem of this equation under the condition $ M(u,v)<M(\phi ,\psi) $, where $ M(u,v) $ denotes the mass and $ (\phi ,\psi) $ is a ground state. In the mass-resonance case, we prove scattering by the argument of Dodson [5]. Scattering is also obtained without mass-resonance condition under the restriction that $ (u,v) $ is radially symmetric.