  We consider the nonlinear stability of spectrally stable periodic waves in
the Lugiato-Lefever equation (LLE), a damped nonlinear Schr\"odinger equation
with forcing that arises in nonlinear optics. So far, nonlinear stability of
such solutions has only been established against co-periodic perturbations by
exploiting the existence of a spectral gap. In this paper, we consider
perturbations which are localized, i.e., integrable on the line. Such localized
perturbations naturally yield the absence of a spectral gap, so we must rely on
a substantially different method with origins in the stability analysis of
periodic waves in reaction-diffusion systems. The relevant linear estimates
have been obtained in recent work by the first three authors through a delicate
decomposition of the associated linearized solution operator. Since its most
critical part just decays diffusively, the nonlinear iteration can only be
closed if one allows for a spatio-temporal phase modulation. However, the
modulated perturbation satisfies a quasilinear equation yielding an apparent
loss of regularity. To overcome this obstacle, we incorporate tame estimates on
the unmodulated perturbation, which satisfies a semilinear equation in which no
derivatives are lost, yet where decay is too slow to close an independent
iteration scheme. We obtain nonlinear stability of periodic steady waves in the
LLE against localized perturbations with precisely the same decay rates as
predicted by the linear theory.
