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In nonlinear optics $2\pi$-periodic solutions $a\in C^2([0,2\pi];\mathbb{C})$of the stationary Lugiato-Lefever equation $-d a"= ({\rm i} -\zeta)a+|a|^2a-{\rm i} f$ serve as a model for frequency combs, which are opticalsignals consisting of a superposition of modes with equally spaced frequencies.We prove that nontrivial frequency combs can only be observed for specialranges of values of the forcing and detuning parameters $f$ and $\zeta$, as ithas been previously documented in experiments and numerical simulations. E.g.,if the detuning parameter $\zeta$ is too large then nontrivial frequency combsdo not exist, cf. Theorem 2. Additionally, we show that for large ranges ofparameter values nontrivial frequency combs may be found on continua whichbifurcate from curves of trivial frequency combs. Our results rely on the proofof a priori bounds for the stationary Lugiato-Lefever equation as well as adetailed rigorous bifurcation analysis based on the bifurcation theorems ofCrandall-Rabinowitz and Rabinowitz. We use the software packages AUTO andMATLAB to illustrate our results by numerical computations of bifurcationdiagrams and of selected solutions.

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