LAUSR

We consider the Lugiato-Lefever model of optical fibers in the periodic context. Spectrally stable periodic steady states were constructed recently in the studies of Delcey and Haragus [Philos. Trans. R. Soc., A 376, 20170188 (2018)]; [Rev. Roumaine Math. Pures Appl. (to be published)]; and Hakkaev et al. (e-print arXiv:1806.04821). The spectrum of the linearization around such solitons consists of simple eigenvalues 0, −2α < 0, while the rest of it is a subset of the vertical line { μ : R μ = − α } . Assuming such a property abstractly, we show that the linearized operator generates a C0 semigroup and, more importantly, the semigroup obeys (optimal) exponential decay estimates. Our approach is based on the Gearhart-Pruss theorem, where the required resolvent estimates may be of independent interest. These results are applied to the proof of asymptotic stability with phase of the steady states.We consider the Lugiato-Lefever model of optical fibers in the periodic context. Spectrally stable periodic steady states were constructed recently in the studies of Delcey and Haragus [Philos. Trans. R. Soc., A 376, 20170188 (2018)]; [Rev. Roumaine Math. Pures Appl. (to be published)]; and Hakkaev et al. (e-print arXiv:1806.04821). The spectrum of the linearization around such solitons consists of simple eigenvalues 0, −2α < 0, while the rest of it is a subset of the vertical line { μ : R μ = − α } . Assuming such a property abstractly, we show that the linearized operator generates a C0 semigroup and, more importantly, the semigroup obeys (optimal) exponential decay estimates. Our approach is based on the Gearhart-Pruss theorem, where the required resolvent estimates may be of independent interest. These results are applied to the proof of asymptotic stability with phase of the steady states.