  We study nonlinear waves on a plane-wave background in an erbium-doped fiber
system, which is governed by the coupled nonlinear Schr\"odinger and the
Maxwell-Bloch equations. We find that prolific different types of nonlinear
localized and periodic waves do exist in the system, including multi-peak
soliton, periodic wave, antidark soliton, and W-shaped soliton (as well as the
known bright soliton, breather, and rogue wave). In particular, the dynamics of
these waves can be extracted from a unified exact solution, and the
corresponding existence conditions are presented explicitly. Our results
demonstrate the structural diversity of the nonlinear waves in this system.
