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

Abstract

AbstractWe consider Kerr frequency combs in a dual-pumped microresonator as time-periodic and spatially $$2\pi $$ 2 π -periodic traveling wave solutions of a variant of the Lugiato–Lefever equation, which is a damped, detuned and driven nonlinear Schrödinger equation given by $$\textrm{i}a_\tau =(\zeta -\textrm{i})a-d a_{x x}-|a|^2a+\textrm{i}f_0+\textrm{i}f_1\textrm{e}^{\textrm{i}(k_1 x-\nu _1 \tau )}$$ i a τ = ( ζ - i ) a - d a xx - | a | 2 a + i f 0 + i f 1 e i ( k 1 x - ν 1 τ ) . The main new feature of the problem is the specific form of the source term $$f_0+f_1\textrm{e}^{\textrm{i}(k_1 x-\nu _1 \tau )}$$ f 0 + f 1 e i ( k 1 x - ν 1 τ ) which describes the simultaneous pumping of two different modes with mode indices $$k_0=0$$ k 0 = 0 and $$k_1\in \mathbb {N}$$ k 1 ∈ 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., $$f_1=0$$ f 1 = 0 , can be continued into the range $$f_1\not =0$$ f 1 ≠ 0 . Our analytical findings apply both for anomalous ($$d>0$$ d > 0 ) and normal ($$d<0$$ d < 0 ) dispersion, and they are illustrated by numerical simulations.