推广的Lugiato-Lefever方程的钉扎
Pinning in the extended Lugiato-Lefever equation
摘要：
我们考虑Lugiato-Lefever方程（LLE）的一个变体，它是一维环面上的非线性薛定谔方程，带有强迫和阻尼，我们在其中加入一个带有势$εV(x)$的一阶导数项.势打破了LLE的平移不变性。根据有效势$V_\text{eff}$，这是一个适当的加权和集成版本的$V$的零的存在，我们表明，从$ε=0$固定的解决方案可以继续局部范围$ε\not =0$。而且，$ε$ -连续解的极值点位于$V_\text{eff}$的零点附近.因此，我们把这种现象称为稳态解的\emph{pinning}。如果我们另外假设在$ε=0$的起始定态解是谱稳定的，由于平移不变性是虚轴上唯一的特征值，因此具有简单的零特征值，我们可以证明其$ε$ -延拓的渐近稳定性或不稳定性取决于$V_\text{eff}'$在$V_\text{eff}$的零处的符号和$ε$的符号。LLE的变体出现在克尔非线性环形微谐振器中的光学频率梳的描述中，该谐振器由两个不同频率和不同功率的不同连续单色光源泵浦。我们的分析结果说明了数值模拟。
英文摘要：
We consider a variant of the Lugiato-Lefever equation (LLE), which is a nonlinear Schrödinger equation on a one-dimensional torus with forcing and damping, to which we add a first-order derivative term with a potential $εV(x)$. The potential breaks the translation invariance of LLE. Depending on the existence of zeroes of the effective potential $V_\text{eff}$, which is a suitably weighted and integrated version of $V$, we show that stationary solutions from $ε=0$ can be continued locally into the range $ε\not =0$. Moreover, the extremal points of the $ε$-continued solutions are located near zeros of $V_\text{eff}$. We therefore call this phenomenon \emph{pinning} of stationary solutions. If we assume additionally that the starting stationary solution at $ε=0$ is spectrally stable with the simple zero eigenvalue due to translation invariance being the only eigenvalue on the imaginary axis, we can prove asymptotic stability or instability of its $ε$-continuation depending on the sign of $V_\text{eff}'$ at the zero of $V_\text{eff}$ and the sign of $ε$. The variant of the LLE arises in the description of optical frequency combs in a Kerr nonlinear ring-shaped microresonator which is pumped by two different continuous monochromatic light sources of different frequencies and different powers. Our analytical findings are illustrated by numerical simulations.
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