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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 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.
@article{AIHPC_2023__40_4_769_0,
author = {Haragus, Mariana and Johnson, Mathew A. and Perkins, Wesley R. and de Rijk, Bj\"orn},
title = {Nonlinear modulational dynamics of spectrally stable {Lugiato{\textendash}Lefever} periodic waves},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
pages = {769--802},
volume = {40},
number = {4},
year = {2023},
doi = {10.4171/aihpc/65},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4171/aihpc/65/}
}
TY - JOUR AU - Haragus, Mariana AU - Johnson, Mathew A. AU - Perkins, Wesley R. AU - de Rijk, Björn TI - Nonlinear modulational dynamics of spectrally stable Lugiato–Lefever periodic waves JO - Annales de l'I.H.P. Analyse non linéaire PY - 2023 SP - 769 EP - 802 VL - 40 IS - 4 UR - http://geodesic.mathdoc.fr/articles/10.4171/aihpc/65/ DO - 10.4171/aihpc/65 LA - en ID - AIHPC_2023__40_4_769_0 ER -
%0 Journal Article %A Haragus, Mariana %A Johnson, Mathew A. %A Perkins, Wesley R. %A de Rijk, Björn %T Nonlinear modulational dynamics of spectrally stable Lugiato–Lefever periodic waves %J Annales de l'I.H.P. Analyse non linéaire %D 2023 %P 769-802 %V 40 %N 4 %U http://geodesic.mathdoc.fr/articles/10.4171/aihpc/65/ %R 10.4171/aihpc/65 %G en %F AIHPC_2023__40_4_769_0
Haragus, Mariana; Johnson, Mathew A.; Perkins, Wesley R.; de Rijk, Björn. Nonlinear modulational dynamics of spectrally stable Lugiato–Lefever periodic waves. Annales de l'I.H.P. Analyse non linéaire, Tome 40 (2023) no. 4, pp. 769-802. doi: 10.4171/aihpc/65
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