Stability of solitary waves for the vector nonlinear Schrِdinger equation in higher-order Sobolev spaces

In this manuscript, a sharp form of orbital stability for the Schrödinger system, also known as the vector NLS, i ∂ ∂ t u j + ∂ 2 ∂ x x u j + 2 ∑ i = 1 m | u i | 2 u j = 0 , where u j are complex-valued functions of ( x , t ) ∈ R 2 , j = 1 , 2 , … , m , is established in L 2 -based Sobolev classes of arbitrarily high order. The result means practically that not only does the bulk of what emanates from the perturbed solitary wave stay close in shape and propagation and phase speeds to the original solitary wave, but emerging residual oscillations must also be very small and not only in the energy norm.