A Priori Bounds and Global Bifurcation Results for Frequency Combs Modeled by the Lugiato-Lefever Equation
A Priori Bounds and Global Bifurcation Results for Frequency Combs Modeled by the Lugiato-Lefever Equation
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DOI：
10.1137/16m1066221
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发表时间：
2016-03
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通讯作者：
Rainer Mandel;W. Reichel
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作者：
Rainer Mandel;W. Reichel
在非线性光学中，定常Lugiato-Lefever方程$-da”=（{\rmi}-\zeta）a +的C^2（[0，2\pi];\mathbb{C}）$中的2\pi $-周期解$a\|一|^2a-{\rm i} f$用作频率梳的模型，频率梳是由频率相等的模式叠加组成的光学信号。我们证明，非平凡的频率梳只能观察到的强迫和失谐参数$f$和$\zeta$的值的特殊范围内，因为它已经被记录在实验和数值模拟。例如，在一个示例中，如果失谐参数太大，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AUTO和MATLAB通过对分叉图和选定解的数值计算来说明我们的结果。
In nonlinear optics $2\pi$-periodic solutions $a\in C^2([0,2\pi];\mathbb{C})$ of the stationary Lugiato-Lefever equation $-d a"= ({\rm i} -\zeta)a +|a|^2a-{\rm i} f$ 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 $\zeta$, as it has been previously documented in experiments and numerical simulations. E.g., if the detuning parameter $\zeta$ 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.