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 Schödinger equations \begin{align*} \begin{cases} i\partial_t u +Δu =\bar{u}v,\\ i\partial_t v +κΔv =u^2, \end{cases} (t,x)\in \mathbb{R}\times \mathbb{R}^4, \end{align*} 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 $M(u,v)<M(\varphi ,\psi )$, where $M(u,v)$ denotes the mass and $(\varphi ,\psi )$ is a ground state. In the mass-resonance case, we prove scattering by the argument of Dodson \cite{MR3406535}. Scattering is also obtained without mass-resonance condition under the restriction that $(u,v)$ is radially symmetric.

49 pages