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On a Schrödinger system arizing in nonlinear optics

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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.

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This work was developed in the frame of the CAPES-FCT convenium Equações de evolução dispersivas. Ademir Pastor would like to thank the kind hospitality of Instituto Superior Técnico, Universidade de Lisboa. Filipe Oliveira was partially supported by the Project CEMAPRE/REM - UIDB/05069/2020 and PTDC/MAT-PUR/1788/2020 financed by FCT/MCTES through national funds. Ademir Pastor was partially supported by CNPq/Brazil Grants 402849/2016-7 and 303762/2019-5.

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Oliveira, F., Pastor, A. On a Schrödinger system arizing in nonlinear optics. Anal.Math.Phys. 11, 123 (2021).

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