谱稳定Lugiato-Lefever周期波的线性调制和次谐波动力学
Linear Modulational and Subharmonic Dynamics of Spectrally Stable Lugiato-Lefever Periodic Waves
摘要：
研究了Lugiato-Lefever方程（LLE）的谱稳定$T$ -周期定态解的线性动力学，LLE是一个在非线性光学中产生的带强迫的阻尼非线性薛定谔方程.这样的$T$ -周期解对于每个$N\in\mathbb{N}$的$NT$ -周期（即次谐波）扰动是非线性稳定的，扰动的指数衰减率为$e^{-δ_N t}$ .然而，指数衰减率$δ_N$和初始扰动的允许大小都倾向于$0$作为$N\to\infty$，因此该结果在$N$中是不均匀的，并且实际上在极限$N=\infty$中是空的。本文的主要目标是介绍一种方法，在LLE的上下文中，通过该方法可以实现至少在线性水平上的次谐波扰动的一致稳定性结果。所得到的均匀衰减率的多项式衰减率的本地化，即在实数线上可积，扰动的LLE的这种光谱稳定的周期解，精确地同意。这项工作既统一和扩大了现有的几个作品在文献中有关的稳定性和动力学等波，并提出了一个一般的方法来研究这些问题在其他情况下。
英文摘要：
We study the linear dynamics of spectrally stable $T$-periodic stationary solutions of the Lugiato-Lefever equation (LLE), a damped nonlinear Schrödinger equation with forcing that arises in nonlinear optics. Such $T$-periodic solutions are nonlinearly stable to $NT$-periodic, i.e. subharmonic, perturbations for each $N\in\mathbb{N}$ with exponential decay rates of perturbations of the form $e^{-δ_N t}$. However, both the exponential rates of decay $δ_N$ and the allowable size of the initial perturbations tend to $0$ as $N\to\infty$, so that this result is non-uniform in $N$ and, in fact, empty in the limit $N=\infty$. The primary goal of this paper is to introduce a methodology, in the context of the LLE, by which a uniform stability result for subharmonic perturbations may be achieved, at least at the linear level. The obtained uniform decay rates are shown to agree precisely with the polynomial decay rates of localized, i.e. integrable on the real line, perturbations of such spectrally stable periodic solutions of the LLE. This work both unifies and expands on several existing works in the literature concerning the stability and dynamics of such waves, and sets forth a general methodology for studying such problems in other contexts.
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