On a Schrödinger system arizing in nonlinear optics

2018/10/18 by Oliveira, Filipe, Pastor, Ademir · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1810.08231

Abstract

We study the nonlinear Schrödinger system \begincases iu_t+Δu-u+((1)/(9)|u|²+2|w|²)u+(1)/(3)u²w=0,
i σw_t+Δw-μw+(9|w|²+2|u|²)w+(1)/(9)u³=0, \endcases for (x,t)∈ ℝⁿ×ℝ, 1≤ n≤ 3 and σ,μ>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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