Nonlinear subharmonic dynamics of spectrally stable Lugiato–Lefever periodic waves

We study the nonlinear dynamics of perturbed, spectrally stable T -periodic stationary solutions of the Lugiato-Lefever equation (LLE), a damped nonlinear Schrödinger equation with forcing that arises in nonlinear optics. It is known that for each $N\in\mathbb{N}$, such a $T$-periodic wave train is asymptotically stable against $NT$-periodic, i.e. subharmonic, perturbations, in the sense that initially nearby data will converge at an exponential rate to a (small) spatial translation of the underlying wave. Unfortunately, in such results both the allowable size of initial perturbations as well as the exponential rates of decay depend on $N$ and, in fact, tend to zero as $N\to\infty$, leading to a lack of uniformity in the period of the perturbation. In recent work, the authors performed a delicate decomposition of the associated linearized solution operator and obtained linear estimates which are uniform in $N$. The dynamical description suggested by this uniform linear theory indicates that the corresponding nonlinear iteration can only be closed if one allows for a spatio-temporal phase modulation of the underlying wave. However, such a modulated perturbation is readily seen to satisfy a quasilinear equation, yielding an inherent loss of regularity. ... mehrWe regain regularity by transferring a nonlinear damping estimate, which has recently been obtained for the LLE in the case of localized perturbations to the case of subharmonic perturbations. Thus, we obtain a nonlinear, subharmonic stability result for periodic stationary solutions of the LLE that is uniform in $N$. This in turn yields an improved nonuniform subharmonic stability result providing an $N$-independent ball of initial perturbations which eventually exhibit exponential decay at an $N$-dependent rate. Finally, we argue that our results connect in the limit $N\to\infty$ previously established stability results against localized perturbations, thereby unifying existing theories.