On a system of nonlinear Schrödinger equations with quadratic interaction

Abstract

We study a system of nonlinear Schrödinger equations with quadratic interaction in space dimension $n⩽6$. The Cauchy problem is studied in $L^2$, in $H^1$, and in the weighted $L^2$ space $\langle x{\rangle }^{-1}{L}^2=\mathcal{F}(H^1)$ under mass resonance condition, where $\langle x\rangle =(1+|x{|}^2{)}^{1/2}$ and $\mathcal{F}$ is the Fourier transform. The existence of ground states is studied by variational methods. Blow-up solutions are presented in an explicit form in terms of ground states under mass resonance condition, which ensures the invariance of the system under pseudo-conformal transformations.