Asymptotic stability for spectrally stable Lugiato-Lefever solitons in periodic waveguides

Abstract

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:Rμ=−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-Prüss 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.

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

“…We also expect that our method could substantially improve nonlinear stability results for periodic waves of the LLE (and for semilinear equations more generally) to subharmonic perturbations, i.e., N T -periodic perturbations with N ∈ N. For example, we point out in the case of the LLE that current techniques [30] exploit the presence of a spectral gap yielding stability results, which are not uniform in N , since the exponential decay rate and the allowable size of initial perturbations are both controlled by the size of the spectral gap which tends to zero as N → ∞; see [11] and also [16]. We aim, through an extension of the stability theory for localized perturbations presented in this paper, to establish a nonlinear stability result against subharmonic perturbations, which is uniform in N ; see forthcoming work [12].…”

“…Theorem 1.3 is the first nonlinear stability result for T -periodic steady waves in the LLE (1.1) against localized perturbations. So far, nonlinear (in)stability of such solutions has only been established against co-periodic perturbations with the aid of standard orbital stability techniques, which exploit the presence of a spectral gap and lead to exponential decay of the perturbed solution to a time-dependent phase modulation of the periodic wave; see [5,23,24,30]. 3 In contrast, Theorem 1.3 establishes algebraic decay of the perturbed solution to a spatio-temporal phase modulation of the underlying wave.…”

Bandwidth and conversion efficiency analysis of dissipative Kerr soliton frequency combs based on bifurcation theory

“…So far, stationary states of the LLE have been mainly investigated by local bifurcation analysis [14,15,[18][19][20][21][22][23], focusing on states in the vicinity of the trivial LLE solution that consists of a single CW tone at the pumped resonance. Global aspects in particular concerning the snaking behavior of solution branches are discussed in [15,19,20], and recently a rigorous stability analysis of stationary states closing the gap between linearized stability and nonlinear stability was achieved in [24]. These methods revealed a large variety of comb states, and were partially extended via numerical continuation methods to regions further away from the trivial state where solitons occur.…”

Linear Modulational and Subharmonic Dynamics of Spectrally Stable Lugiato-Lefever Periodic Waves

“…For the periodic waves bifurcating from the dnoidal solutions of the NLS equation, the authors proved in [10] that some of these are spectrally stable to co-periodic perturbations. That this spectral stability corresponds to nonlinear stability was recently established in [28,Theorem 1]. The power of this result is that it reduces the problem of nonlinear stability for co-periodic perturbations to a spectral problem which, in turn, may be amenable to well-conditioned analytical and numerical methods.…”