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arXiv · 1703.10509

On a nonlinear Schrödinger system arising in quadratic media

Adán Corcho·Simão Correia·Filipe Oliveira·Jorge D. Silva
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Abstract

We consider the quadratic Schrödinger system $$iu_t+Δ_{γ_1}u+\overline{u}v=0$$ $$2iv_t+Δ_{γ_2}v-βv+\frac 12 u^2=0,$$ where $t\in\mathbf{R},\,x\in \mathbf{R}^d\times \mathbf{R}$, in dimensions $1\leq d\leq 4$ and for $γ_1,γ_2>0$, the so-called elliptic-elliptic case. We show the formation of singularities and blow-up in the $L^2$-(super)critical case. Furthermore, we derive several stability results concerning the ground state solutions of this system.

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Adán Corcho, Simão Correia, Filipe Oliveira, Jorge D. Silva. 2017-07-20. On a nonlinear Schrödinger system arising in quadratic media. https://arxiv.org/abs/1703.10509

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