论文标题
通过两种模式抽水获得的频率梳子的Lugiato-Lefever模型的解决方案的全局连续性
Global continua of solutions to the Lugiato-Lefever model for frequency combs obtained by two-mode pumping
论文作者
论文摘要
我们将双泵的微孔子中的kerr频率梳理为时间周期和空间上的$2π$ - 周期性的行驶波解决方案的变体，它是$ \ rm {i} a __- A_ {X X} - | A |^2a+\ Mathrm {I} F_0+\ MathRM {I} F_1 \ Mathrm {E}^{\ Mathrm {i}（k_1 x-nmrm {k_1 x-v_1τ）} $。 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 $k_1\in \mathbb{N}$.我们证明了基于A-Priori界限和固定点定理的这些行进波的存在和唯一定理。此外，通过使用隐式函数定理和分叉理论，我们可以展示如何从$ 1 $ -MODE CASE（即$ f_1 = 0 $）中进行非分类解决方案，可以继续介于$ f_1 \ f_1 \ not = 0 $。我们的分析结果适用于异常（$ d> 0 $）和普通（$ d <0 $）分散，并通过数值模拟进行了说明。
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.
