Lugiato-Lefever方程频率梳的先验界和全局分支结果
A priori bounds and global bifurcation results for frequency combs modeled by the Lugiato-Lefever equation
摘要：
在非线性光学中，稳态Lugiato-Lefever方程$2π$的周期解$a\in C^2([0,2π];\mathbb{C})$用作频率梳的模型，频率梳是由具有相等间隔频率的模式的叠加组成的光信号。我们证明，非平凡的频率梳只能观察到的强迫和失谐参数$f$和$ζ$的值的特殊范围内，因为它已经在以前的实验和数值模拟记录。例如，在一个示例中，如果失谐参数$ζ$太大，则不存在非平凡的频率梳，参见。定理2.此外，我们表明，对于大范围的参数值的非平凡的频率梳可能会发现连续分叉平凡的频率梳的曲线。我们的研究结果依赖于固定的Lugiato-Lefever方程的先验界的证明以及基于Crandall-Rabinowitz和Rabinowitz分歧定理的详细严格的分歧分析。我们使用软件包MATLAB和MATLAB来说明我们的结果通过数值计算的分歧图和选定的解决方案。
英文摘要：
In nonlinear optics $2π$-periodic solutions $a\in C^2([0,2π];\mathbb{C})$ of the stationary Lugiato-Lefever equation $-d a"= ({\rm i} -ζ)a +|a|^2a-{\rm i} f$ serve as a model for frequency combs, which are optical signals consisting of a superposition of modes with equally spaced frequencies. We prove that nontrivial frequency combs can only be observed for special ranges of values of the forcing and detuning parameters $f$ and $ζ$, as it has been previously documented in experiments and numerical simulations. E.g., if the detuning parameter $ζ$ is too large then nontrivial frequency combs do not exist, cf. Theorem 2. Additionally, we show that for large ranges of parameter values nontrivial frequency combs may be found on continua which bifurcate from curves of trivial frequency combs. Our results rely on the proof of a priori bounds for the stationary Lugiato-Lefever equation as well as a detailed rigorous bifurcation analysis based on the bifurcation theorems of Crandall-Rabinowitz and Rabinowitz. We use the software packages AUTO and MATLAB to illustrate our results by numerical computations of bifurcation diagrams and of selected solutions.
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