We study the nonlinear Schrödinger system

$$\begin{aligned} \left\{ \begin{array}{lllll} \displaystyle iu_t+\Delta u-u+\left( \frac{1}{9}|u|^2+2|w|^2\right) u+\frac{1}{3}{\overline{u}}^2w=0,\\ i\displaystyle \sigma w_t+\Delta w-\mu w+(9|w|^2+2|u|^2)w+\frac{1}{9}u^3=0, \end{array}\right. \end{aligned}$$

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.

中文翻译：

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非线性光学中的薛定谔系统

我们研究非线性薛定谔系统

$$\begin{对齐} \left\{ \begin{array}{llll} \displaystyle iu_t+\Delta u-u+\left( \frac{1}{9}|u|^2+2|w|^2 \right) u+\frac{1}{3}{\overline{u}}^2w=0,\\ i\displaystyle \sigma w_t+\Delta w-\mu w+(9|w|^2+2|u |^2)w+\frac{1}{9}u^3=0, \end{array}\right. \end{对齐}$$

对于\((x,t)\in {\mathbb {R}}^n\times {\mathbb {R}}\) , \(1\le n\le 3\)和\(\sigma ,\mu >0\)。该系统模拟了具有克尔型非线性响应的材料中光束与其三次谐波之间的相互作用。我们证明了基态解的存在，分析了它的稳定性，并建立了局部和全局适定结果以及几个爆炸标准。