Bifurcation structure of periodic patterns in the Lugiato-Lefever equation with anomalous dispersion
Bifurcation structure of periodic patterns in the Lugiato-Lefever equation with anomalous dispersion
复制标题
反常色散Lugiato-Lefever方程中周期模式的分叉结构
作者：
P. Parra;D. Gomila;L. Gelens;E. Knobloch
本文研究了Lugiato-Lefever方程所描述的具有kerr型非线性和异常群速度色散的非线性光学谐振腔中空间扩展图样的稳定性和分岔结构。虽然存在不同波长的单参数图案族，但我们关注的是由均匀态的调制不稳定性引起的具有临界波数${k}_{c}$的图案。我们发现与此模式相关的解分支与波数$2{k}_{c}$的模式分支相连。下一个分支也连接到双波数的模式分支，这次是$4{k}_{c}$，这个过程通过一系列2:1的空间共振重复。当失谐参数在临界波数${k}_{c}$以下趋于零时，失谐参数趋于$ensuremath{heta}=2$，这种分岔结构与组织空间局域亮孤子的叶状蛇形分岔结构有关。这些模式经历的二次分岔和由此产生的时间动力学也进行了研究。
We study the stability and bifurcation structure of spatially extended patterns arising in nonlinear optical resonators with a Kerr-type nonlinearity and anomalous group velocity dispersion, as described by the Lugiato-Lefever equation. While there exists a one-parameter family of patterns with different wavelengths, we focus our attention on the pattern with critical wave number ${k}_{c}$ arising from the modulational instability of the homogeneous state. We find that the branch of solutions associated with this pattern connects to a branch of patterns with wave number $2{k}_{c}$. This next branch also connects to a branch of patterns with double wave number, this time $4{k}_{c}$, and this process repeats through a series of 2:1 spatial resonances. For values of the detuning parameter approaching $ensuremath{ heta}=2$ from below the critical wave number ${k}_{c}$ approaches zero and this bifurcation structure is related to the foliated snaking bifurcation structure organizing spatially localized bright solitons. Secondary bifurcations that these patterns undergo and the resulting temporal dynamics are also studied.