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

Adán J. Corcho (Instituto de Matemática, Universidade Federal do Rio de Janeiro, Brazil)

Simão Correia (Center for Mathematics, Fundamental Applications and Operations Research and Faculdade de Ciências, Universidade de Lisboa, Portugal)

Filipe Oliveira (Mathematics Department and CEMAPRE. ISEG, Universidade de Lisboa, Portugal)

Jorge D. Silva (Center for Mathematical Analysis, Geometry and Dynamical Systems, Department of Mathematics, Instituto Superior Técnico, Universidade de Lisboa, Portugal)

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

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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