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 $$2\pi $$ 2 π -periodic traveling wave solutions of variant the Lugiato–Lefever equation, which is damped, detuned driven nonlinear Schrödinger equation given by $$\textrm{i}a_\tau =(\zeta -\textrm{i})a-d a_{x x}-|a|^2a+\textrm{i}f_0+\textrm{i}f_1\textrm{e}^{\textrm{i}(k_1 x-\nu _1 \tau )}$$ i a τ = ( ζ - ) d xx | + f 0 1 e k x ν . The main new feature problem specific form source term $$f_0+f_1\textrm{e}^{\textrm{i}(k_1 describes simultaneous pumping two different modes with mode indices $$k_0=0$$ $$k_1\in \mathbb {N}$$ ∈ N prove existence uniqueness theorems for these waves based on a-priori bounds fixed point theorems. Moreover, using implicit function theorem bifurcation theory, we show how non-degenerate from 1-mode case, i.e., $$f_1=0$$ , can be continued into range $$f_1\not =0$$ ≠ Our analytical findings apply both anomalous ( $$d>0$$ > ) normal $$d<0$$ < dispersion, they are illustrated numerical simulations.