  We first report the first- and higher-order vector Peregrine solitons (alias
rational rogue waves) for the any multi-component NLS equations based on the
loop group theory, an explicit (n + 1)-multiple eigenvalue of a characteristic
polynomial of degree (n + 1) related to the condition of Benjamin-Feir
instability, and inverse functions. Particularly, these vector rational rogue
waves are parity-time symmetric for some parameter constraints. A systematic
and effective approach is proposed to study the asymptotic behaviors of these
vector rogue waves such that the decompositions of rogue waves are related to
the so-called governing polynomials, which pave a powerful way in the study of
vector rogue wave structures of the multi-component integrable systems. The
vector rogue waves with maximal amplitudes can be determined via the parameter
vectors, which is interesting and useful in the multi-component physical
systems.
