Adiabatic perturbation theory for the vector nonlinear Schrödinger equation with nonvanishing boundary conditions

We consider a defocusing Manakov system (vector nonlinear Schrödinger (NLS) system) with nonvanishing boundary conditions and use the inverse scattering transform formalism. Integrable models provide a very useful proving ground for testing new analytic and numerical approaches to studying the vector NLS system. We develop a perturbation theory for the integrable vector NLS model. Evidently, small disturbance of the integrability condition can be considered a perturbation of the integrable model. Our formalism is based on the Riemann–Hilbert problem associated with the vector NLS model with nonvanishing boundary conditions. We use the RH and adiabatic perturbation theory to analyze the dynamics of dark–dark and dark–bright solitons in the presence of a perturbation with nonvanishing boundary conditions.