Global continua of solutions to the Lugiato–Lefever model for frequency combs obtained by two-mode pumping

Abstract

We consider Kerr frequency combs in a dual-pumped microresonator as time-periodic and spatially 2pi -periodic traveling wave solutions of a variant of the Lugiato–Lefever equation, which is a damped, detuned and driven nonlinear Schrödinger equation given by textrm{i}a_tau =(zeta -textrm{i})a-d a_{x x}-|a|^2a+textrm{i}f_0+textrm{i}f_1textrm{e}^{textrm{i}(k_1 x-nu _1 tau )}. The main new feature of the problem is the specific form of the source term f_0+f_1textrm{e}^{textrm{i}(k_1 x-nu _1 tau )} which describes the simultaneous pumping of two different modes with mode indices k_0=0 and k_1in 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_1not =0. Our analytical findings apply both for anomalous (d>0) and normal (d<0) dispersion, and they are illustrated by numerical simulations.