Nonlinear optics plays a crucial part in the progress of laser-based technologies and optical science. In this paper, we investigate the three-coupled variable-coefficient nonlinear Schrödinger system, which describes the amplification or attenuation of the picosecond pulses in an inhomogeneous multicomponent optical fiber with different frequencies or polarizations. Based on the existing Lax pair, we construct the first-/second-order generalized Darboux transformations and obtain the second-order semirational rogue-wave solutions, which represent the slowly varying envelopes of optical modes, under a constraint among the fiber gain/loss, nonlinearity and group velocity dispersion. We obtain the influences of nonlinearity and group velocity dispersion: when the value of the nonlinearity increases, amplitudes of the second-order semirational rogue waves decrease and when the value of the group velocity dispersion increases, amplitudes of the second-order semirational rogue waves increase. Baseband modulation instability (MI) through the linear stability explanation is obtained. When the characteristic roots have the negative imaginary parts, the system appears the baseband MI. When the MI occurs, it is of baseband type. With the positive parts, however, there is no MI occurring.

中文翻译：

---

非均匀多分量光纤中三耦合变系数非线性薛定谔系统的广义达布变换、半有理流氓波和调制不稳定性

非线性光学在基于激光的技术和光学科学的进步中起着至关重要的作用。在本文中，我们研究了三耦合可变系数非线性薛定谔系统，该系统描述了皮秒脉冲在具有不同频率或偏振的非均匀多分量光纤中的放大或衰减。基于现有的Lax对，我们构造一阶/二阶广义Darboux变换，得到二阶半有理流氓波解，它代表了在光纤增益/损耗之间的约束下光模的慢变包络，非线性和群速度色散。我们得到了非线性和群速度色散的影响：当非线性值增加时，二阶半有理流氓波的幅度减小，当群速度色散值增加时，二阶半有理流氓波的幅度增加。基带调制不稳定性（MI）通过线性稳定性解释得到。当特征根具有负虚部时，系统出现基带MI。当 MI 发生时，它是基带类型的。然而，对于积极的部分，没有 MI 发生。它是基带类型的。然而，对于积极的部分，没有 MI 发生。它是基带类型的。然而，对于积极的部分，没有 MI 发生。