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 $2\pi$-periodic traveling wave solutions of a variant of the Lugiato-Lefever equation, which is a damped, detuned and driven nonlinear Schrödinger equation given by $ia_\tau = (\zeta − i ) a − da_{xx} − |a|^2 a + i f_0 + i f_1\text{e}^{i( k_1 x−ν_1 \tau)}$. The main new feature of the problem is the specific form of the source term $f_0 + f_1 \text{e}^{i(k_1 x− ν_1 \tau )}$ 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 $f_1\ne 0$. Our analytical findings apply both for anomalous $(d > 0)$ and normal $(d < 0)$ dispersion, and they are illustrated by numerical simulations.