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

In this paper, consideration is given to the 3-coupled nonlinear Schrodinger system i ∂ ∂ t u j + ∂ 2 ∂ x 2 u j + ∑ i = 1 3 b i j | u i | 2 u j = 0 , where u j are complex-valued functions of ( x , t ) ∈ R 2 , j = 1 , 2 , 3 , and b i j are positive constants satisfying b i j = b j i . It will be shown first that if the symmetric matrix B = ( b i j ) satisfies certain conditions, then ground-state solutions of the 3-coupled nonlinear Schrodinger 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 Schrodinger system was considered.