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Research has shown that passively mode-locked fiber lasers produce chaotic output, which has caught the attention of physicists, chemists, and bio-scientists owing to their wide bandwidth, good random characteristics, and strong anti-interference. In passively mode-locked fiber lasers, soliton pulsations and soliton explosions with period bifurcation characteristics have been demonstrated to be effective paths to chaos as far as 20 years ago. However, due to the lack of real-time spectrum measurement techniques, the earlier research investigated their theoretical aspect. In recent years, the rise of the dispersive Fourier transform technique has activated an upsurge of their experimental research. The present work first discussed the theoretical model of passively mode-locked fiber lasers, the computational analysis method of soliton dynamics, and the related theory of dispersive Fourier transform technique. In addition, we presented and evaluated the progress of the theoretical and experimental research on soliton pulsations as well as on soliton explosions in passively mode-locked fiber lasers. Finally, we proposed the future research directions of the soliton pulsations and soliton explosions that offer great promise for scientific discoveries.

We have experimentally investigated a novel kind of soliton molecule in a graphene-based mode-locking fiber laser with anomalous dispersion. The soliton molecule exhibits a stable rectangular profile on the oscilloscope, whereas it shows randomly distributed peaks in the autocorrelation trace, which indicates that the temporal separation of pulses in the molecule is varying all the time. The optical spectrum is modulated with a depth of about 7 dB over the whole profile, induced by the interaction of the intra-molecule solitons. The experimental results demonstrate that solitons in the pulse molecule oscillate randomly in temporal domain and the neighboring molecules are temporally separated by a fundamental cavity repetition rate, which is very different from that of the multi-solitons or bound-state solitons.

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