  We study multi-peak solitons \textit{on a plane-wave background} in an
erbium-doped fiber system with some higher-order effects, which is governed by
a coupled Hirota and Maxwel-Bloch (H-MB) model. The important characteristics
of multi-peak solitons induced by the higher-order effects, such as the
velocity changes, localization or periodicity attenuation, and state
transitions, are revealed in detail. In particular, our results demonstrate
explicitly that a multi-peak soliton can be converted to an anti-dark soliton
when the periodicity vanishes; on the other hand, a multi-peak soliton is
transformed to a periodic wave when the localization vanishes. Numerical
simulations are performed to confirm the propagation stability of multi-peak
solitons riding on a plane-wave background. Finally, we compare and discuss the
similarity and difference of multi-peak solitons in special degenerate cases of
the H-MB system with general existence conditions.
