Nonlinear Subharmonic Dynamics of Spectrally Stable Lugiato–Lefever Periodic Waves

Abstract

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 \(N\rightarrow \infty \), 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. We 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 \(N \rightarrow \infty \) to previously established stability results against localized perturbations, thereby unifying existing theories.