Orbital stability of solitary waves of a 3-coupled nonlinear Schrödinger system

In this paper, consideration is given to the 3-coupled nonlinear Schrödinger system i∂∂tuj+∂2∂x2uj+∑i=13bij|ui|2uj=0, where ujuj are complex-valued functions of (x,t)∈R2(x,t)∈R2, j=1,2,3j=1,2,3, and bijbij are positive constants satisfying bij=bjibij=bji. It will be shown first that if the symmetric matrix B=(bij)B=(bij) satisfies certain conditions, then ground-state solutions of the 3-coupled nonlinear Schrödinger system exist, and moreover, they are orbitally stable. The theory is then extended to include solitary waves as well. In particular, it will be shown that when a solitary wave is perturbed, the perturbed solution must stay close to a solitary-wave profile in which the translation and phase parameters are prescribed functions of time. Properties of these functions are then studied. This is a continuous work of our previous paper where the 2-coupled nonlinear Schrödinger system was considered.