Linear Modulational and Subharmonic Dynamics of Spectrally Stable Lugiato-Lefever Periodic Waves

Abstract: We study the linear dynamics of spectrally stable $T$-periodic stationary solutions of the Lugiato-Lefever equation (LLE), a damped nonlinear Schr\"odinger equation with forcing that arises in nonlinear optics. Such $T$-periodic solutions are nonlinearly stable to $NT$-periodic, i.e. subharmonic, perturbations for each $N\in\mathbb{N}$ with exponential decay rates of perturbations of the form $e^{-\delta_N t}$. However, both the exponential rates of decay $\delta_N$ and the allowable size of the initial perturbations tend to $0$ as $N\to\infty$, so that this result is non-uniform in $N$ and, in fact, empty in the limit $N=\infty$. The primary goal of this paper is to introduce a methodology, in the context of the LLE, by which a uniform stability result for subharmonic perturbations may be achieved, at least at the linear level. The obtained uniform decay rates are shown to agree precisely with the polynomial decay rates of localized, i.e. integrable on the real line, perturbations of such spectrally stable periodic solutions of the LLE. This work both unifies and expands on several existing works in the literature concerning the stability and dynamics of such waves, and sets forth a general methodology for studying such problems in other contexts.