Study provides mathematical proof of nonlinear stability in spectrally stable periodic waves

The Lugiato-Lefever equation, derived over 30 years ago, provides a model to describe spontaneous pattern formation in the field of nonlinear fiber optics. It describes the generation of Kerr frequency combs, series of equally spaced, sharp spectral lines generated from a continuous wave pump laser by the Kerr effect, a nonlinear effect often exploited in optical systems. Kerr frequency combs have numerous potential applications in high-capacity telecommunications, chemical sensing, astronomy and more.

In their article, Stanislavova and Stefanov investigate the asymptotic stability of spectrally stable periodic solutions of the Lugiato-Lefever equation. The existence of such solutions was recently established, which is of significant practical interest since they correspond to stable Kerr frequency combs. The authors wanted to confirm the longtime behavior of these steady states by delving into a rigorous mathematical proof.

They employed a powerful abstract tool from the theory of operator semigroups called the Gearhart-Prüss theorem, which converted an intimidating problem into a manageable one. Stanislavova and Stefanov were then able to solve the reformulated problem using harmonic analysis and some operator theory methods in a fairly efficient way.

In the end, the authors showed that spectrally stable periodic waves for the Lugiato-Lefever model are nonlinearly stable. This result wasn’t entirely surprising to them, since most features of the model have been already observed by physicists and simulated numerically. Nonetheless, their contribution consists of the rigorous proof that describes the behavior of the system. The work has the potential to highlight new features of the model that haven’t been observed yet and to possibly suggest new experiments that may be conducted.

Source: “Asymptotic stability for spectrally stable Lugiato-Lefever solitons in periodic waveguides,” by Milena Stanislavova and Atanas G. Stefanov, Journal of Mathematical Physics (2018). The article can be accessed at https://doi.org/10.1063/1.5048017 .