On a Schrödinger system arizing in nonlinear optics

Abstract

We study the nonlinear Schrödinger system \[ \begin{cases} \displaystyle iu_t+Δu-u+(\frac{1}{9}|u|^2+2|w|^2)u+\frac{1}{3}\overline{u}^2w=0,\\ i\displaystyle σw_t+Δw-μw+(9|w|^2+2|u|^2)w+\frac{1}{9}u^3=0, \end{cases} \] for $(x,t)\in {\mathbb{R}}^n\times \mathbb{R}$, $1\le n\le 3$ 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.