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

Chuangye Liu, Nghiem V. Nguyen & Zhi-Qiang Wang

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 $i\frac{∂}{∂ t}u_j+\frac{∂²}{∂x²}u_j+Σ^m_{i=1}b_{ij}|u_i|^p|u_j|^{p-2}u_j=0$, 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 asminimizers 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

  • AMS Subject Headings

35A15, 35B35, 35Q35

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

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 = {Liu , Chuangye and Nguyen , Nghiem V. and Wang , Zhi-Qiang}, 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 $i\frac{∂}{∂ t}u_j+\frac{∂²}{∂x²}u_j+Σ^m_{i=1}b_{ij}|u_i|^p|u_j|^{p-2}u_j=0$, 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 asminimizers 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 KW - coupled NLS systems KW - vector solutions KW - ground-state solutions AB -

In this paper, the existence and stability results for ground state solutions of an m-coupled nonlinear Schrödinger system $i\frac{∂}{∂ t}u_j+\frac{∂²}{∂x²}u_j+Σ^m_{i=1}b_{ij}|u_i|^p|u_j|^{p-2}u_j=0$, 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 asminimizers 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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