| Abstract: |
| We consider the Lugiato-Lefever (LL) model of optical fibers. We construct a two parameter family of steady state solutions, i.e. Kerr frequency combs, for small pumping parameter and the correspondingly (and necessarily) small detuning parameter. We identify the spectrally stable ones, and moreover we have a fairly explicit description of the spectrum of the linearized operator. We show that these waves are asymptotically stable by phase. The main ingredient of the proof is that the associated semi-group has strictly negative growth rate, away from the translational eigenvalue at zero. This is obtained as a consequence of delicate resolvent estimates and an application of the Gearhart-Pruss theorem. |
|