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# Communications in Mathematical Sciences

## Volume 17 (2019)

### Number 4

### On a nonlinear Schrödinger system arising in quadratic media

Pages: 969 – 987

DOI: https://dx.doi.org/10.4310/CMS.2019.v17.n4.a5

#### Authors

#### Abstract

We consider the quadratic Schrödinger system\begin{cases}iu_t + \Delta_{\gamma_1} u + \bar{u}v = 0 \\2iv_t + \Delta_{\gamma_2} v - \beta v + \frac{1}{2} u^2 = 0 , &t \in \mathbb{R} , x \in \mathbb{R}^d \times \mathbb{R} ,\end{cases}in dimensions $1 \leq d \leq 4$ and for $\gamma_1 ,\gamma_2 \gt 0$, 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.

#### Keywords

nonlinear Schrödinger systems, blow-up, ground states, stability

#### 2010 Mathematics Subject Classification

35C08, 35Q55, 35Q60

Simão Correia was partially supported by Fundação para a Ciência e Tecnologia, through the grant SFRH/BD/96399/2013 and through contract UID/MAT/04561/2013. Filipe Oliveira was partially supported by the Project CEMAPRE - UID/ MULTI/00491/2013 financed by FCT/MCTES through national funds. Jorge D. Silva was partially supported by FCT/Portugal through UID/MAT/04459/2013.

Received 17 October 2018

Accepted 30 January 2019

Published 25 October 2019