Existence and stability of a two-parameter family of solitary waves for a 2-coupled nonlinear Schrödinger system

Abstract

In this paper, the existence and stability results for a two-parameter family of vector solitary-wave solutions (i.e both components are nonzero) of the 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 $a,b,c \in \mathbb R$are established. The results extend our earlier ones as well as those of Ohta, Cipolatti and Zumpichiatti and de Figueiredo and Lopes. As opposed to other methods used before to establish existence and stability where the two constraints of the minimization problems are related to each other, our approach here characterizes solitary-wave solutions as minimizers of an energy functional subject to two independent constraints. The set of minimizers is shown to be stable; and depending on the interplay between the parameters $a,b$ and $c$, further information about the structures of this set are given.