We consider a coherently coupled nonlinear Schr\"odinger equation with modulated self-phase modulation, cross-phase modulation, and four-wave mixing nonlinearities and varying refractive index in anisotropic graded index nonlinear medium. By identifying an appropriate similarity transformation, we obtain a general localized wave solution and investigate their dynamics with a proper set of modulated nonlinearities. In particular, our study reveals different manifestations of localized waves such as stable solitons, Akhmediev breathers, Ma breathers, and rogue waves of bright, bright-dark, and dark-dark type and explores their manipulation mechanism with suitably engineered nonlinearity parameters. We have provided a categorical analysis with adequate graphical demonstrations.

中文翻译：

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在具有四波混合效应的耦合非线性 Schrõdinger 系统中设计光学杂散波和呼吸器

我们考虑在各向异性渐变折射率非线性介质中具有调制自相位调制、交叉相位调制和四波混合非线性和可变折射率的相干耦合非线性 Schr\"odinger 方程。通过确定适当的相似变换，我们获得了一般局部波解并用一组适当的调制非线性研究它们的动力学。特别是，我们的研究揭示了局部波的不同表现形式，例如稳定孤子、Akhmediev 呼吸器、Ma 呼吸器和明、明-暗和暗的流氓波-dark 类型并使用适当设计的非线性参数探索它们的操纵机制。我们提供了具有足够图形演示的分类分析。