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

TL;DR

This paper establishes bounds and bifurcation structures for frequency comb solutions modeled by the Lugiato-Lefever equation, linking parameter ranges to the existence of nontrivial solutions in nonlinear optics.

Contribution

It provides the first rigorous bounds and bifurcation analysis for frequency combs in the Lugiato-Lefever model, confirming experimental and numerical observations.

Findings

Nontrivial frequency combs exist only within specific parameter ranges.

Large detuning parameters prevent the existence of nontrivial combs.

Bifurcation from trivial solutions leads to nontrivial combs on continuous branches.

Abstract

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 optical signals consisting of a superposition of modes with equally spaced frequencies. We prove that nontrivial frequency combs can only be observed for special ranges of values of the forcing and detuning parameters $f$ and $\zeta$, as it has been previously documented in experiments and numerical simulations. E.g., if the detuning parameter $\zeta$ 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. We use the software packages AUTO and MATLAB to illustrate our results by numerical computations of bifurcation diagrams and of selected solutions.