Scattering for the quadratic NLS system in ${\mathbb{R}}^5$ without mass resonance or radial symmetry

Abstract

We consider the quadratic NLS system \begin{equation*} \begin{cases} i\partial_tu+Δu=v\bar{u},\newline i\partial_tv+κΔv=u^2, \end{cases} \qquad (t,x)\in\mathbb{R}\times\mathbb{R}^5, \end{equation*} where $\kappa >0$. If $\kappa =1/2$, which is called mass resonance condition, then scattering below the ground state is shown by Hamano. Moreover, when $\kappa \ne 1/2$, scattering of radial solutions is proved in Hamano--Inui--Nishimura (2021). In the present paper, we prove scattering below the ground state for $\kappa \ne 1/2$ without radial symmetry. Our proof is based on the concentration compactness and rigidity method by Kenig--Merle (2006). For the rigidity argument, following Pausader (2010), we use a localized virial argument in the direction orthogonal to the momentum.