Dissipative soliton molecules with independently evolving or flipping phases in mode-locked fiber lasers

We numerically demonstrate the existence of a discrete family of robust dissipative soliton bound state solutions (soliton molecules) in a mode-locked fiber laser with an instantaneous saturable absorber in the normal dispersion domain. For a certain domain of the small-signal gain, we obtain a robust first-level bound state with almost constant separation where the phase of the two pulses evolves independently. Moreover, their phase difference can evolve either periodically or chaotically depending on the small-signal gain. Interestingly, higher level bound states exhibit a fundamentally different dynamics. They represent oscillating solutions with a phase difference alternating between zero and $\ensuremath{\pi}$. We identify the crucial role of the linear gain saturation for the existence of these robust molecules independently of their level.