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The spatially periodic breather solutions (SPBs) of the nonlinearSchr\"odinger equation, prominent in modeling rogue waves, are unstable. Inthis paper we numerically investigate the effects of nonlinear dissipation andhigher order nonlinearities on the routes to stability of the SPBs in theframework of the nonlinear damped higher order nonlinear Schr\"odinger(NLD-HONLS) equation. The initial data used in the experiments are generated byevaluating exact SPB solutions at time $T_0$. The number of instabilities ofthe background Stokes wave and the damping strength are varied. The Floquetspectral theory of the NLS equation is used to interpret and provide acharacterization of the perturbed dynamics in terms of nearby solutions of theNLS equation. Significantly, as $T_0$ is varied, tiny bands of complex spectrumare observed to pinch off in the Floquet decomposition of the NLD-HONLS data,reflecting the breakup of the SPB into a waveform that is close to either a oneor two "soliton-like" structure. For wide ranges of $T_0$, i.e. for solutionsinitialized in the early to middle stage of the development of the MI, allrogue waves are observed to occur when the spectrum is close to a one or twosoliton-like state. When the solutions are initialized as the MI is saturating,rogue waves also can occur after the spectrum has left a soliton-like state.Other novel features arise due to nonlinear damping: enhanced asymmetry, twotimescales in the evolution of the spectrum and a delay in the growth ofinstabilities due to frequency downshifting.

Nonlinear damped spatially periodic breathers and the emergence of soliton-like rogue waves