A priori bounds and global bifurcation results for frequency combs modeled by the Lugiato-Lefever equation

In nonlinear optics 2Pi-periodic solutions a 2 C2([0; 2Pi];C) of the stationary Lugiato-Lefever equation -da00 = (i-)a + jaj2a - if serve as a model for frequency combs, which are optical signals consisting of a superposition of modes with equally spaced frequencies. In accordance with experimental data we prove that nontrivial frequency combs can only be observed for special values of the forcing and detuning parameters f, . E.g., if the detuning parameter is too large then nontrivial frequency combs do not exist, cf. Theorem 2. Additionally, we show that for large ranges of parameter values nontrivial frequency combs may be found on continua which bifurcate from curves of trivial frequency combs. Our results rely on the proof of a priori bounds for the stationary Lugiato-Lefever equation as well as a detailed rigorous bifurcation analysis based on the bifurcation theorems of Crandall-Rabinowitz and Rabinowitz.