通过双模泵浦获得的频率梳的 Lugiato-Lefever 模型解的全局连续统
Global continua of solutions to the Lugiato-Lefever model for frequency combs obtained by two-mode pumping
AI总结：
本文研究了双泵浦微环谐振腔中克尔频率梳对应的 Lugiato-Lefever 方程变体的行波解，基于先验界和不动点定理证明了解的存在性与唯一性，并利用隐函数定理和分岔理论将单模解延拓至双模范围。
摘要：
我们将双泵浦微环谐振腔中的克尔频率梳视为 Lugiato-Lefever 方程变体的时间周期且空间 $2π$-周期的行波解，该方程是一个阻尼、失谐且驱动的非线性薛定谔方程，由 $\mathrm{i}a_τ=(ζ-\mathrm{i})a - d a_{x x}-|a|^2a+\mathrm{i}f_0+\mathrm{i}f_1\mathrm{e}^{\mathrm{i}(k_1 x-ν_1 τ)}$ 给出。该问题的主要新特征是源项 $f_0+f_1\mathrm{e}^{\mathrm{i}(k_1 x-ν_1 τ)}$ 的特定形式，它描述了模式指标为 $k_0=0$ 和 $k_1\in \mathbb{N}$ 的两个不同模式的同时泵浦。我们基于先验界和不动点定理证明了这些行波解的存在性和唯一性定理。此外，通过使用隐函数定理和分岔理论，我们展示了如何将 $1$ 模情况（即 $f_1=0$）下的非退化解延拓至 $f_1\not =0$ 的范围。我们的解析结果适用于反常（$d>0$）和正常（$d<0$）色散，并通过数值模拟进行了说明。
英文摘要：
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.
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