Global continua of solutions to the Lugiato-Lefever model for frequency combs obtained by two-mode pumping
We consider Kerr frequency combs in a dual-pumped microresonator as time-periodic and spatially $2π$-periodic traveling wave solutions of a variant of the Lugiato-Lefever equation, which is a damped, detuned and driven nonlinear Schrödinger equation given by $\mathrm{i}a_τ=(ζ-\mathrm{i})a - d a_{x x}-|a|^2a+\mathrm{i}f_0+\mathrm{i}f_1\mathrm{e}^{\mathrm{i}(k_1 x-ν_1 τ)}$. The main new feature of the problem is the specific form of the source term $f_0+f_1\mathrm{e}^{\mathrm{i}(k_1 x-ν_1 τ)}$ which describes the simultaneous pumping of two different modes with mode indices $k_0=0$ and $k_1\in \mathbb{N}$. We prove existence and uniqueness theorems for these traveling waves based on a-priori bounds and fixed point theorems. Moreover, by using the implicit function theorem and bifurcation theory, we show how non-degenerate solutions from the $1$-mode case, i.e. $f_1=0$, can be continued into the range $f_1\not =0$. Our analytical findings apply both for anomalous ($d>0$) and normal ($d<0$) dispersion, and they are illustrated by numerical simulations.