Volume 49, Issue 2
Existence and Stability of Solitary Waves of an M-Coupled Nonlinear Schrödinger System

J. Math. Study, 49 (2016), pp. 132-148.

Published online: 2016-07

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• Abstract

In this paper, the existence and stability results for ground state solutions of an m-coupled nonlinear Schrödinger system $$i\frac{∂}{∂ t}u_j+\frac{∂²}{∂x²}u_j+\sum\limits^m_{i=1}b_{ij}|u_i|^p|u_j|^{p-2}u_j=0,$$ are established, where $2 ≤ m, 2≤p<3$ and $u_j$ are complex-valued functions of $(x,t) ∈ \mathbb{R}^2, j=1,...,m$ and $b_{ij}$ are positive constants satisfying $b_{ij}=b_{ji}$. In contrast with other methods used before to establish existence and stability of solitary wave solutions where the constraints of the variational minimization problem are related to one another, our approach here characterizes ground state solutions as minimizers of an energy functional subject to independent constraints. The set of minimizers is shown to be orbitally stable and further information about the structure of the set is given in certain cases.

• Keywords

Orbital stability, coupled NLS systems, vector solutions, ground-state solutions.

35A15, 35B35, 35Q35

chuangyeliu@mail.ccnu.edu.cn (Chuangye Liu)

nghiem.nguyen@usu.edu (Nghiem V. Nguyen)

zhi-qiang.wang@usu.edu (Zhi-Qiang Wang)

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@Article{JMS-49-132, author = {Chuangye and Liu and chuangyeliu@mail.ccnu.edu.cn and 13317 and School of Mathematics and Statistics and Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan 430079, Hubei, P.R. China and Chuangye Liu and Nghiem V. and Nguyen and nghiem.nguyen@usu.edu and 13318 and Department of Mathematics and Statistics, Utah State University, Logan, UT 84322, USA and Nghiem V. Nguyen and Zhi-Qiang and Wang and zhi-qiang.wang@usu.edu and 12244 and Center for Applied Mathematics, Tianjin University, Tianjin 300072, P.R. China, and Department of Mathematics and Statistics, Utah State University, Logan, UT 84322, USA and Zhi-Qiang Wang}, title = {Existence and Stability of Solitary Waves of an M-Coupled Nonlinear Schrödinger System}, journal = {Journal of Mathematical Study}, year = {2016}, volume = {49}, number = {2}, pages = {132--148}, abstract = {

In this paper, the existence and stability results for ground state solutions of an m-coupled nonlinear Schrödinger system $$i\frac{∂}{∂ t}u_j+\frac{∂²}{∂x²}u_j+\sum\limits^m_{i=1}b_{ij}|u_i|^p|u_j|^{p-2}u_j=0,$$ are established, where $2 ≤ m, 2≤p<3$ and $u_j$ are complex-valued functions of $(x,t) ∈ \mathbb{R}^2, j=1,...,m$ and $b_{ij}$ are positive constants satisfying $b_{ij}=b_{ji}$. In contrast with other methods used before to establish existence and stability of solitary wave solutions where the constraints of the variational minimization problem are related to one another, our approach here characterizes ground state solutions as minimizers of an energy functional subject to independent constraints. The set of minimizers is shown to be orbitally stable and further information about the structure of the set is given in certain cases.

}, issn = {2617-8702}, doi = {https://doi.org/10.4208/jms.v49n2.16.03}, url = {http://global-sci.org/intro/article_detail/jms/995.html} }
TY - JOUR T1 - Existence and Stability of Solitary Waves of an M-Coupled Nonlinear Schrödinger System AU - Liu , Chuangye AU - Nguyen , Nghiem V. AU - Wang , Zhi-Qiang JO - Journal of Mathematical Study VL - 2 SP - 132 EP - 148 PY - 2016 DA - 2016/07 SN - 49 DO - http://doi.org/10.4208/jms.v49n2.16.03 UR - https://global-sci.org/intro/article_detail/jms/995.html KW - Orbital stability, coupled NLS systems, vector solutions, ground-state solutions. AB -

In this paper, the existence and stability results for ground state solutions of an m-coupled nonlinear Schrödinger system $$i\frac{∂}{∂ t}u_j+\frac{∂²}{∂x²}u_j+\sum\limits^m_{i=1}b_{ij}|u_i|^p|u_j|^{p-2}u_j=0,$$ are established, where $2 ≤ m, 2≤p<3$ and $u_j$ are complex-valued functions of $(x,t) ∈ \mathbb{R}^2, j=1,...,m$ and $b_{ij}$ are positive constants satisfying $b_{ij}=b_{ji}$. In contrast with other methods used before to establish existence and stability of solitary wave solutions where the constraints of the variational minimization problem are related to one another, our approach here characterizes ground state solutions as minimizers of an energy functional subject to independent constraints. The set of minimizers is shown to be orbitally stable and further information about the structure of the set is given in certain cases.

Chuangye Liu, Nghiem V. Nguyen & Zhi-Qiang Wang. (2019). Existence and Stability of Solitary Waves of an M-Coupled Nonlinear Schrödinger System. Journal of Mathematical Study. 49 (2). 132-148. doi:10.4208/jms.v49n2.16.03
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