AI 中文总结
该研究对带周期驱动的阻尼非线性薛定谔方程的亮、暗脉冲进行严格分析，构造出对应实验宽带频率梳的周期多脉冲解，为Lugiato-Lefever方程提供了首个从黑NLS孤子出发的严格分叉结果。
AI 中文摘要
我们分析了带有周期驱动的阻尼非线性薛定谔（NLS）方程中亮脉冲与暗脉冲的存在性和稳定性。该Lugiato-Lefever方程是非线性光学中的经典模型，用于描述由双色激光泵浦驱动的耗散克尔腔中的斑图形成。我们从亮NLS孤子和黑NLS孤子分叉出发，严格构造了Lugiato-Lefever方程的亮单脉冲与暗单脉冲解。利用新近开发的空间周期系统中脉冲解的拼接与周期延拓工具箱，我们得到了对应于实验观测到的宽带频率梳的周期多脉冲解。亮脉冲的存在性与稳定性可由标准的Lyapunov-Schmidt约化和Krein指数论证得到，而暗脉冲的构造则更为精细，因为黑NLS孤子的非零渐近态会导致线性化算子出现中性本质谱。通过沿分叉路径仔细追踪相关的小空间Floquet指数并运用指数二分法，我们确立了暗脉冲解的存在性，这类解由两个畴壁和一个具有临界绝对谱的长平台连接而成。据我们所知，这是首次从黑NLS孤子出发对Lugiato-Lefever方程得到严格分叉结果。
英文摘要
We analyze the existence and stability of bright and dark pulses in a damped nonlinear Schrödinger (NLS) equation with periodic forcing. This Lugiato-Lefever equation is a canonical model in nonlinear optics describing pattern formation in a dissipative Kerr cavity driven by a bichromatic laser pump. Bifurcating from the bright and black NLS solitons, we rigorously construct bright and dark single-pulse solutions to the Lugiato-Lefever equation. Using a recently developed toolbox for concatenating and periodically extending pulse solutions in spatially periodic systems, we then obtain periodic multipulse solutions corresponding to experimentally observed broad-bandwidth frequency combs. While the existence and stability of the bright pulses follow from standard Lyapunov-Schmidt reduction and Krein index arguments, the construction of the dark pulses is substantially more delicate because the nonzero asymptotic states of the black NLS soliton yield neutral essential spectrum of the linearization. By carefully tracking the associated small spatial Floquet exponents along the bifurcation and employing exponential dichotomies, we establish the existence of dark pulse solutions. These consist of two domain walls connected by a long plateau with critical absolute spectrum. To our knowledge, this constitutes the first rigorous bifurcation result in the Lugiato-Lefever equation from the black NLS soliton.
