Modulational instability of the Kuznetsov-Ma breather in optical fibers with constant and periodic dispersion

•Performed a time-dependent modulational instability (MI) analysis of the Kuznetsov-Ma (KM) breather.•Studied MI of the KM breather in terms of time and provided a theoretical prediction for the bifurcation.•Studied MI of the KM breather in terms of an oscillating dispersion and discussed possible parametric resonance.•Discussed the experimental feasibility of observing the resonance.

We perform modulational instability analysis of the Kuznetsov-Ma breather (KMB) propagating in a medium with constant and periodic dispersions. In the constant dispersion case, we show that the most unstable modulational instability eigenmode leads to bifurcation of the KMB into a sequence of breathers. The time at which bifurcation starts and the bifurcation rate are quantitatively accounted for. In the case of a KMB propagating in an optical fiber with a constant dispersion perturbed by a sinusoidal time-dependent modulation our results suggest that the KMB is most unstable just before and after the breather reaches its maximum peak height. Parametric resonance occurs when the frequency of the dispersion modulation is much larger than the natural frequency of the breather.