Orbital stability of solitary waves for a nonlinear Schrödinger system

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 $b^2> ac$.