On the multicomponent nonlinear Schrödinger equation in the case of non-vanishing boundary conditions

The inverse scattering transform is applied to the analysis of the multicomponent non-linear Schrödinger equation in the class of $s\times s$ matrix-valued functions $q(x)$, $\lim_{x\to\pm\infty}q(x)=q_\pm$, $q_+q_+^+=q_-q_-^+$. A number of peculiarities, due to the non-vanishing boundary conditions are exhibited, including: i) the existence of several chemical potentials; ii) the modifications in the integrals of motion and the Poisson brackets between the scattering data.