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In an interesting recent analysis, Haragus-Johnson-Perkins-de Rijk have shownmodulational stability under localized perturbations of steady periodicsolutions of the Lugiato-Lefever equation (LLE), in the process pointing out adifficulty in obtaining standard "nonlinear damping estimates" on modulatedperturbation variables to control regularity of solutions. Here, we point outthat in place of standard "inverse-modulated" damping estimates, one canalternatively carry out a damping estimate on the "forward-modulated"perturbation, noting that norms of forward- and inverse-modulated variables areequivalent modulo absorbable errors, thus recovering the classical argumentstructure of Johnson-Noble-Rodrigues-Zumbrun for parabolic systems. Thisobservation seems of general use in situations of delicate regularity.  Applied in the context of (LLE) it gives the stronger result of stability andasymptotic behavior with respect to nonlocalized perturbations.

Forward-modulated damping estimates and nonlocalized stability of periodic Lugiato-Lefever wave