Stability of normalized solitary waves for three coupled nonlinear Schrödinger equations

Abstract

In this paper we establish existence and stability results concerning fullynontrivial solitary-wave solutions to 3-coupled nonlinear Schrödinger system\begin{equation*} i\partial_t u_{j}+\partial_{xx}u_{j}+ \left(\sum_{k=1}^{3} a_{kj} |u_k|^{p}\right)|u_j|^{p-2}u_j = 0, \ j=1,2,3,\end{equation*}where $u_j$ are complex-valued functions of $(x,t)\in \mathbb{R}^{2}$ and $a_{kj}$ arepositive constants satisfying $a_{kj}=a_{jk}$ (symmetric attractive case).Our approach improves many of the previously known results. In all variational methods used previouslyto study the stability of solitary waves,which we are aware of, the constraint functionals were not independently chosen.Here we study a problem of minimizing the energy functional subject tothree independent $L^2$ mass constraints andestablish existence and stability results fora true three-parameter family of solitary waves.