On the delay of lag synchronization in breathing soliton molecules

Synchronization is a fundamental cornerstone of nonlinear science, particularly in the realm of nonlinear optics where it governs the coherent interaction of complex dissipative structures. As localized temporal structures with periodic oscillatory behavior, breathing solitons provide an ideal platform for investigating these synchronization phenomena. While the synchronization of single breathers has been extensively studied, the internal synchronization within breathing soliton molecules (BSMs) remains largely unexplored. In this work, we investigate the synchronous evolution of BSMs in passively mode-locked fiber lasers and reveal the presence of internal lag synchronization within these dissipative structures. Our results demonstrate that while the synchronization delays in diatomic BSMs are relatively small and can be directly observed, triatomic BSMs exhibit much larger delays that is difficult to determine. We employ a cross-correlation-based method to quantify these substantial delays. The triatomic BSMs provide an additional degree of freedom for synchronization regulation compared to diatomic ones. Notably, we find that non-adjacent pulses within a molecule can achieve zero-lag synchronization.