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.
Categories: math.AP