Existence and Stability of Solitary Waves of an M-Coupled Nonlinear Schrödinger System

Abstract

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