Linear modulational and subharmonic dynamics of spectrally stable Lugiato-Lefever periodic waves

Abstract

We consider the nonlinear stability of spectrally stable periodic waves in the Lugiato-Lefever equation (LLE), a damped nonlinear Schrödinger 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 inherent loss of regularity which, due to the low-order damping in the LLE, cannot be regained using standard methods. Therefore, in contrast to the case of reaction-diffusion systems, the nonlinear iteration cannot be closed for the modulated perturbation. In this work, we present a new nonlinear iteration scheme incorporating 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.

Subharmonic dynamics of wave trains in the Korteweg‐de Vries/Kuramoto‐Sivashinsky equation

“…This methodology was first introduced, at the linear level, by the authors and collaborators in Ref. 24 and was further extended, in the context of reaction-diffusion systems, to the full nonlinear level in Ref. 25, and is closely modeled off of the known stability theory for localized perturbations.…”

“…For more details, see the recent works. 24,25 Suppose that ū is a 𝑇-periodic traveling wave solution of (1) with wave speed 𝑐 ∈ ℝ, and consider the associated linearized operator . For each fixed 𝑁 ∈ ℕ, define b…”

Subharmonic Dynamics of Wave Trains in the Korteweg-de Vries / Kuramoto-Sivashinsky Equation

“…We begin by reviewing the results of Floquet-Bloch theory when applied to subharmonic perturbations. For more details, see the recent works [8,12]. Suppose that ū is a T -periodic traveling wave solution of (1.1) with wave speed c ∈ R, and consider the associated linearized operator L. For each fixed N ∈ N, define 2 for some ξ ∈ Ω N and w ∈ L 2 per (0, T ).…”

“…Naturally, the dimension of this total eigenspace is tending to infinity as N → ∞, and we must work to establish uniform in N decay estimates associated with the induced decomposition of the semigroup. This methodology was first introduced, at the linear level, by the authors and collaborators in [8] and was further extended, in the context of reactiondiffusion systems, to the full nonlinear level in [12], and is closely modeled off of the known stability theory for localized perturbations [10]. In the present work, the additional complication (compared to these previous subharmonic works) is the presence of a non-trivial Jordan block (1.4) which, in turn, yields slower uniform decay rates of the associated linear semigroups as compared to those rates in the reaction-diffusion context.…”

Subharmonic Dynamics of Wave Trains in Reaction Diffusion Systems

“…Note that, due to Lemma 2.5 we expect the "critical frequency" component to be dominated by the translational mode φ ′ . This decomposition was recently carried out in detail (in a related context) in [4], and for completeness we review it here. Note the decomposition is heavily motivated by the corresponding decomposition used in the case of localized perturbations: see [8,6].…”