On a Schr\"odinger system arizing in nonlinear optics

TL;DR

This paper investigates a coupled nonlinear Schr"odinger system modeling optical interactions, establishing existence, stability, well-posedness, and blow-up criteria for solutions in dimensions one to three.

Contribution

It provides the first rigorous analysis of ground states, stability, and well-posedness for this specific Schr"odinger system arising in nonlinear optics.

Findings

Existence of ground state solutions confirmed.

Stability analysis of the ground states conducted.

Local and global well-posedness established, along with blow-up criteria.

Abstract

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.