Stability analysis of dark pulse solutions

Establish the stability of the dark pulse solutions to the Lugiato–Lefever equation constructed in the normal-dispersion regime with periodic forcing, including the spectral behavior associated with the long periodic plateau.

Background

The paper proves existence and nondegeneracy of dark primary pulses and constructs dark frequency combs, but it does not analyze their stability. The anticipated difficulty comes from the critical absolute spectrum of the long periodic plateau state: the origin lies in the plateau’s absolute spectrum, and eigenvalues of the pulse linearization may accumulate there as the plateau length increases. The authors therefore leave the stability problem unresolved and specifically point to the need to track eigenvalues near zero.

References

Theorem~\ref{thm:main_dark} does not address the stability of the dark pulse solutions, which we leave for future work. We expect the stability analysis to be particularly delicate, primarily because the origin belongs to the absolute spectrum associated with the long periodic plateau state connecting the front and back interfaces of the dark primary pulse.

— Dissipative bright and dark pulses under periodic forcing  (2608.14077 - Bengel et al., 14 Aug 2026) in Section 2, Remark following Corollary \ref{cor:dark_FC}; see also Section 2, \S\ref{sec:method_of_proof}