We discuss modulation instability for the generalized nonlinear Schrdinger equation based on nonzero background wave frequency. First of all, we analyze the existence condition of modulation instability under different perturbed frequency. The influences of the background amplitude, background frequency and perturbed frequency on the modulation instability gain are researched, respectively. Also, we obtain the correspondences between several nonlinear excitations (Kuznetsov-Ma breather, general breather, rogue wave, bright soliton and plane wave) and modulation instability according to new parameters. Furthermore, by the Fourier transformation method, we perform spectrum analyses of the first-order and second-order rogue waves. The perturbed frequency of the rogue wave can affect the location and profile of the spectrum. And we find that the spectrum of the second-order rogue wave is jagged due to the collision of the rogue waves. These results would help us further understand the dynamics of rogue wave in complex systems.

我们讨论了基于非零背景波频率的广义非线性 Schrdinger 方程的调制不稳定性。首先，我们分析了不同扰动频率下调制不稳定的存在条件。分别研究了背景幅度、背景频率和扰动频率对调制不稳定增益的影响。此外，我们根据新参数获得了几种非线性激发（Kuznetsov-Ma 呼吸器、一般呼吸器、流氓波、亮孤子和平面波）与调制不稳定性之间的对应关系。此外，通过傅里叶变换方法，我们对一阶和二阶杂波进行了频谱分析。恶意波的扰动频率会影响频谱的位置和轮廓。并且我们发现由于流氓波的碰撞，二阶流氓波的频谱是锯齿状的。这些结果将有助于我们进一步了解复杂系统中流氓波的动力学。