On a nonlinear Schr\"odinger system arising in quadratic media

We consider the quadratic Schr"odinger system $$i{u}_t+{\mathrm{\Delta }}_{{\gamma }_1}u+\bar{u}v=0$$ $$2i{v}_t+{\mathrm{\Delta }}_{{\gamma }_2}v-\beta v+\frac{1}{2}{u}^2=0,$$ where $t\in$\mathbf{R}$,,x\in $\mathbf{R}$^d\times $\mathbf{R}$$,indimensions$1\leq d\leq 4$andfor$\gamma_1,\gamma_2>0$, 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.