The optical rogue wave (RW), known as a short-lived extraordinarily high amplitude dynamics phenomenon with small appearing probabilities, plays an important role in revealing and understanding the fundamental physics of nonlinear wave propagations in optical systems. The random fiber laser (RFL), featured with cavity-free and “modeless” structure, has opened up new avenues for fundamental physics research and potential practical applications combining nonlinear optics and laser physics. Here, the extreme event of optical RW induced by noise-driven modulation instability that interacts with the cascaded stimulated Brillouin scattering, the quasi-phase-matched four-wave mixing as well as the random mode resonance process is observed in a Brillouin random fiber laser comb (BRFLC). Temporal and statistical characteristics of the RWs concerning their emergence and evolution are experimentally explored and analyzed. Specifically, temporally localized structures with high intensities including chair-like pulses with a sharp leading edge followed by a trailing plateau appear frequently in the BRFLC output, which can evolve to chair-like RW pulses with adjustable pulse duration and amplitude under controlled conditions. This investigation provides a deep insight into the extreme event of RWs and paves the way for RW manipulation for its generation and elimination in RFLs through adapted laser configuration.

The objective of this study is to investigate miscellaneous wave structures for perturbed Fokas–Lenells equation (FLE) with cubic-quartic dispersion in polarization-preserving fibers. Based on the improved projective Riccati equations method, various types of soliton solutions such as bright soliton, combo dark–bright soliton, singular soliton and combo singular soliton are constructed. Additionally, a set of periodic singular waves are also retrieved. The dynamical behaviors of some obtained solutions are depicted to provide a key to understanding the physics of the model. The modulation instability of the FLE is reported by employing the linear stability analysis which shows that all solutions are stable.The objective of this study is to investigate miscellaneous wave structures for perturbed Fokas–Lenells equation (FLE) with cubic-quartic dispersion in polarization-preserving fibers. Based on the improved projective Riccati equations method, various types of soliton solutions such as bright soliton, combo dark–bright soliton, singular soliton and combo singular soliton are constructed. Additionally, a set of periodic singular waves are also retrieved. The dynamical behaviors of some obtained solutions are depicted to provide a key to understanding the physics of the model. The modulation instability of the FLE is reported by employing the linear stability analysis which shows that all solutions are stable.