Stability plays a significant role in successfully observing Fermi-Pasta-Ulam-Tsingou (FPUT) phenomenon in experiment. However, there are few relevant studies in the literature. The primary object of this work is to study the stability of FPUT phenomenon in the nonlinear fibers numerically. In this study, we take the sinusoidally perturbed continuous waves (CWs) with white noise as the imposed initial condition, which can be readily realized in real experiments. We find that both the perturbation amplitude and phase difference between the perturbation and pump can drastically affect the stability of the resulting FPUT phenomenon. Firstly, as the perturbation amplitude increases, the FPUT phenomenon becomes much more stable. When the perturbation amplitude reaches a critical value, the most stable FPUT phenomenon can be observed. With the further increase of the perturbation amplitude, the stability of the resulting FPUT phenomenon weakens. Secondly, the phase difference between the perturbation and pump takes distinct effects on the stability of FPUT phenomenon for perturbation frequency located inside and outside the conventional modulation instability (MI) band. For the perturbation frequency located inside the conventional MI band, as the phase difference between the perturbation and pump increases from zero, the corresponding FPUT phenomenon first is less stable, and then becomes most instable at a critical phase difference; after that, the stability of the FPUT phenomenon is enhanced again. For the perturbation frequency located outside the conventional MI band, the stability of FPUT phenomenon is enhanced monotonically as the phase difference increases from 0 to π/2.In order to observe a much more stable FPUT phenomenon, as shown in the above results, the perturbation amplitude should be moderately large, and the phase difference between the perturbation and the pump should be appropriate to avoid the most instable FPUT phenomenon.
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