Effective slow dynamics models for a class of dispersive systems

We consider dispersive systems of the form
∂_t U = Λ_U U + B_U (U, V ), ε∂_t V = Λ_V V + B_V (U, U )
in the singular limit ε → 0, where Λ_U , Λ_V are linear and B_U, B_V bilinear mappings. We are interested in deriving error estimates for the approximation obtained through the regular limit system
∂_t ψ_U = Λ_U ψ_U − B_U (ψ_U , Λ^{−1}_V B_V (ψ_U, ψ_U ))
from a more general point of view. Our abstract approximation theorem applies to a number of semilinear systems, such as the Dirac-Klein-Gordon system, the Klein-Gordon-Zakharov system, and a mean field polaron model. It extracts the common features of scattered results in the literature, but also gains an approximation result for the Dirac-Klein-Gordon system which has not been documented in the literature before. We explain that our abstract approximation theorem is sharp in the sense that there exists a quasilinear system of the same structure where the regular limit system makes wrong predictions.