On a nonlinear Schrödinger system arising in quadratic media

Abstract

We consider the quadratic Schrödinger system $$i{u}_t+{\Delta }_{{\gamma }_1}u+\overline{u}v=0$$ $$2i{v}_t+{\Delta }_{{\gamma }_2}v-\beta v+\frac{1}{2}{u}^2=0,$$ where $t\in \mathbf{R},x\in {\mathbf{R}}^d\times \mathbf{R}$, in dimensions $1\le d\le 4$ and for ${\gamma }_1,{\gamma }_2>0$, the so-called elliptic-elliptic case. We show the formation of singularities and blow-up in the $L^2$-(super)critical case. Furthermore, we derive several stability results concerning the ground state solutions of this system.