Perturbation theory for nearly integrable multicomponent nonlinear PDEs

The Riemann-Hilbert problem associated with the integrable PDE is used as a nonlinear transformation of the nearly integrable PDE to the spectral space. The temporal evolution of the spectral data is derived with account for arbitrary perturbations and is given in the form of exact equations, which generate the sequence of approximate ordinary differential equations in successive orders with respect to the perturbation. For vector nearly integrable PDEs, embracing the vector nonlinear Schroedinger and complex modified Korteweg-de Vries equations, the main result is formulated in a theorem. For a single vector soliton the evolution equations for the soliton parameters and first-order radiation are given in explicit form