On the generation of stable Kerr frequency combs in the Lugiato-Lefever model of periodic optical waveguides

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 $h>0$ and the correspondingly (and necessarily) small detuning parameter, $\alpha >0$. These are $O(1)$ waves, as they are constructed as bifurcation from the standard cnoidal solutions of the cubic NLS. We identify the spectrally stable ones, and more precisely, we show that the spectrum of the linearized operator contains the eigenvalues $0,-2\alpha$, while the rest of it is a subset of $\{\mu :\mathrm{\Re }\mu =-\alpha \}$. This is in line with the expectations for effectively damped Hamiltonian systems, such as the LL model.