Stable standing waves for cubic nonlinear Schrödinger systems with partial confinement

Huifang Jia; Gongbao Li; Xiao Luo

doi:10.3934/dcds.2020148

Article Contents

2020, Volume 40, Issue 5: 2739-2766. Doi: 10.3934/dcds.2020148

This issue Previous Article Movement of time-delayed hot spots in Euclidean space for a degenerate initial state Next Article Measure theoretic pressure and dimension formula for non-ergodic measures

Stable standing waves for cubic nonlinear Schrödinger systems with partial confinement

1.
School of Mathematics and Statistics, Central China Normal University, Wuhan, 430079, China
2.
School of Mathematics and Information Science, Guangzhou University, Guangzhou, 510405, China
3.
School of Mathematics, Hefei University of Technology, Hefei, 230009, China

^* Corresponding author: Gongbao Li
^* Corresponding author: Gongbao Li

Received: June 2019

Published: March 2020

The project is supported by National Natural Science Foundation of China(Grant No.11771166), Hubei Key Laboratory of Mathematical Sciences and Program for Changjiang Scholars and Innovative Research Team in University #IRT_17R46. H.F Jia is also supported by National Postdoctoral Science Foundation of China(Grant No.2019M662833). X. Luo is also supported by National Natural Science Foundation of China(Grant No.11901147)

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Abstract

In this article, we deal with the existence, qualitative and symmetry properties of normalized solutions to the following nonlinear Schrödinger system

$ \begin{equation*} \begin{cases} -\Delta u+(x_{1}^{2}+x_{2}^{2})u = \lambda_{1}u+\mu_{1}u^{3}+\beta uv^{2}, &\quad x\in \mathbb{R}^3,\\ -\Delta v+(x_{1}^{2}+x_{2}^{2})v = \lambda_{2}v+\mu_{2}v^{3}+\beta u^{2}v, &\quad x\in \mathbb{R}^3, \end{cases} \end{equation*} $

where $ \mu_{i}>0 $ ($ i $ = 1, 2), $ \beta>0 $, and the frequencies $ \lambda_{1} $, $ \lambda_{2} $ are unknown and appear as Lagrange multipliers. In addition, we study the stability of the corresponding standing waves for the related time-dependent Schrödinger systems. We mainly extend the results in J. Bellazzini et al. (Commun. Math. Phys. 2017), which dealt with mass-supercritical nonlinear Schrödinger equation with partial confinement, to cubic nonlinear Schrödinger systems with partial confinement.

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Mathematics Subject Classification: Primary: 35J20, 35J25, 35J60.

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