On a Schr\"odinger system arizing in nonlinear optics

We study the nonlinear Schr\"odinger system $$\begin{cases} \displaystyle iu_t+\Delta u-u+(\frac{1}{9}|u|^2+2|w|^2)u+\frac{1}{3}\overline{u}^2w=0,\\ i\displaystyle \sigma w_t+\Delta w-\mu w+(9|w|^2+2|u|^2)w+\frac{1}{9}u^3=0, \end{cases}$$ for $(x,t)\in \mathbb{R}^n\times\mathbb{R}$, $1\leq n\leq 3$ 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.