In this paper we investigate the effect of nonlinear damping on the Lugiato–Lefever equation \(\mathrm {i}\partial _t a = -(\mathrm {i}-\zeta ) a - da_{xx} -(1+\mathrm {i}\kappa )|a|^2a +\mathrm {i}f\) on the torus or the real line. For the case of the torus it is shown that for small nonlinear damping \(\kappa >0\) stationary spatially periodic solutions exist on branches that bifurcate from constant solutions whereas all nonconstant stationary \(2\pi \)-periodic solutions disappear when the damping parameter \(\kappa \) exceeds a critical value. These results apply both for normal (\(d<0\)) and anomalous (\(d>0\)) dispersion. For the case of the real line we show by the Implicit Function Theorem that for small nonlinear damping \(\kappa >0\) and large detuning \(\zeta \gg 1\) and large forcing \(f\gg 1\) strongly localized, bright solitary stationary solutions exist in the case of anomalous dispersion \(d>0\). These results are achieved by using techniques from bifurcation and continuation theory and by proving a convergence result for solutions of the time-dependent Lugiato–Lefever equation.

在本文中，我们研究了非线性阻尼对Lugiato-Lefever方程\（\ mathrm {i} \ partial _t a =-（\ mathrm {i}-\ zeta）a-da_ {xx}-（1+ \圆环或实线上的mathrm {i} \ kappa} | a | ^ 2a + \ mathrm {i} f \）。对于圆环的情况表明，对于小非线性阻尼\（\ kappa> 0 \），固定分支上的常数存在固定的空间周期解，而该分支与恒定解分叉，而所有非恒定的固定\（2 \ pi \）-周期解在阻尼参数\（\ kappa \）超过临界值。这些结果适用于正常（\（d <0 \））和异常（\（d> 0 \））分散。对于实线情况，我们通过隐函数定理表明，对于较小的非线性阻尼\（\ kappa> 0 \）和较大的失谐\（\ zeta \ gg 1 \）和较大的强迫\（f \ gg 1 \）在反常色散\（d> 0 \）的情况下，存在强烈局部的，明亮的孤立平稳解。这些结果是通过使用分叉和延续理论中的技术以及证明与时间相关的Lugiato-Lefever方程解的收敛结果而获得的。