Advances in Differential Equations

Orbital stability of solitary waves for a nonlinear Schrödinger system

Nghiem V. Nguyen and Zhi-Qiang Wang

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Abstract

In this paper, we study the coupled nonlinear Schrödinger system \begin{equation*} \left\{ \begin{matrix} iu_t+ u_{xx} + (a |u|^2 + b |v|^2) u=0\\ iv_t+ v_{xx} + (b |u|^2 + c |v|^2) v=0\\ \end{matrix} \right. \end{equation*} where $u,v$ are complex-valued functions of $(x,t)\in \mathbb R^2$, and $a,b,c \in \mathbb R$. Our work shows that, for this system of equations, the interplay between components of solutions in terms of the parameters $a,b,c$ plays an important role in both the existence and stability of solitary waves. In particular, we prove that solitary wave solutions to this system are orbitally stable when either $0<b < \min\{a,c\}$, or $b>0$ with $b > \max\{a,c\}$ and $b^2> ac$.

Article information

Source
Adv. Differential Equations Volume 16, Number 9/10 (2011), 977-1000.

Dates
First available in Project Euclid: 17 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ade/1355703184

Mathematical Reviews number (MathSciNet)
MR2850761

Subjects
Primary: 35A15: Variational methods 35B35: Stability 35Q35: PDEs in connection with fluid mechanics 35Q35: PDEs in connection with fluid mechanics

Citation

Nguyen, Nghiem V.; Wang, Zhi-Qiang. Orbital stability of solitary waves for a nonlinear Schrödinger system. Adv. Differential Equations 16 (2011), no. 9/10, 977--1000. https://projecteuclid.org/euclid.ade/1355703184.


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