We derive a theoretical model with frequency-dependent parameters for describing the transmission of electromagnetic waves in $\mathcal{PT}$-symmetric metamaterials (MMs). Based on the derived theoretical model, we investigate the tunable band structure and eigenvalue problem in the negative or positive index region of MMs, and predict the existence and stability of fundamental, out-of-phase, and in-phase dipole gap solitons in defocusing or focusing nonlinear $\mathcal{PT}$-symmetric MMs. Furthermore, we discuss the enhanced localization of solitons under different formation conditions. The results reveal that there exist abundant gap solitons with frequency controllability in $\mathcal{PT}$-symmetric MMs, and the established theoretical model paves the way to explore more solitons in $\mathcal{PT}$-symmetric MMs.

中文翻译：

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PT对称非线性超材料中频率相关的基波孤子和偶极子间隙孤子

我们推导了一个具有频率相关参数的理论模型，用于描述电磁波的传输$\mathcal{PT}$-对称超材料（MM）。基于推导的理论模型，我们研究了MM负折射率区域和正折射率区域的可调谐能带结构和特征值问题，并预测了散焦中基波孤子、异相孤子和同相偶极子间隙孤子的存在性和稳定性或聚焦非线性$\mathcal{PT}$-对称MM。此外，我们讨论了不同形成条件下孤子的增强局域化。结果表明，存在丰富的频率可控的带隙孤子。$\mathcal{PT}$-对称MM，所建立的理论模型为探索更多孤子铺平了道路$\mathcal{PT}$-对称MM。