On a Schr\"odinger system arizing in nonlinear optics

We study the nonlinear Schr"odinger system $$\{\begin{array}{l}{\displaystyle i{u}_t+\mathrm{\Delta }u-u+(\frac{1}{9}\mathrm{\mid }u{\mathrm{\mid }}^2+2\mathrm{\mid }w{\mathrm{\mid }}^2)u+\frac{1}{3}{\bar{u}}^{2}w=0,}\\ i{\displaystyle \sigma w_t+\mathrm{\Delta }w-\mu w+(9\mathrm{\mid }w{\mathrm{\mid }}^2+2\mathrm{\mid }u{\mathrm{\mid }}^2)w+\frac{1}{9}{u}^3=0,}\end{array}$$ for $(x,t)\in {\mathbb{R}}^n\times \mathbb{R}$, $1\le n\le 3$ and $\sigma ,\mu >0$. This system models the interaction between an optical beam and its third harmonic in a material with Kerr-type nonlinear response. We prove the existence of ground state solutions, analyse its stability, and establish local and global well-posedness results as well as several criteria for blow-up.