Abstract We show both numerically and experimentally that a dissipative type of Akhmediev-breathers (ABs) and Peregrine solitons (PSs) can be formed in a fiber laser, and their features could be well described by the laser Ginzburg-Landau equation. Moreover, we show that laser gain bandwidth limitation effect can arrest the dissipative ABs. Consequently, a stable one-dimensional periodic pulse train or periodic PS crystal structure can be formed. A kind of movable ‘PSs’ has also been obtained in our fiber laser. The solitons have a PS-like profile but exhibit similar features as those of the dissipative solitons formed in a mode locked fiber laser.