Wave dynamics in topological materials has been widely studied recently. A striking feature is the existence of robust and chiral wave propagations that have potential applications in many fields. A common way to realize such wave patterns is to utilize Dirac points which carry topological indices and is supported by the symmetries of the media. In this work, we investigate these phenomena in photonic media. Starting with Maxwell's equations with a honeycomb material weight as well as the nonlinear Kerr effect, we first prove the existence of Dirac points in the dispersion surfaces of transverse electric and magnetic Maxwell operators under very general assumptions of the material weight. Our assumptions on the material weight are almost the minimal requirements to ensure the existence of Dirac points in a general hexagonal photonic crystal. We then derive the associated wave packet dynamics in the scenario where the honeycomb structure is weakly modulated. It turns out the reduced envelope equation is generally a two-dimensional nonlinear Dirac equation with a spatially varying mass. By studying the reduced envelope equation with a domain-wall-like mass term, we realize the subtle wave motions which are chiral and immune to local defects. The underlying mechanism is the existence of topologically protected linear line modes, also referred to as edge states. However, we show that these robust linear modes do not survive with nonlinearity. We demonstrate the existence of nonlinear line modes, which can propagate in the nonlinear media based on high-accuracy numerical computations. Moreover, we also report a new type of nonlinear modes which are localized in both directions.

中文翻译：

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调制蜂窝介质中的线性和非线性电磁波

最近，拓扑材料中的波动力学得到了广泛的研究。一个显着的特点是存在强大的手征波传播，在许多领域都有潜在的应用。实现这种波浪模式的常见方法是利用狄拉克点，这些点带有拓扑指数并由介质的对称性支持。在这项工作中，我们研究了光子介质中的这些现象。从具有蜂窝材料重量的麦克斯韦方程以及非线性克尔效应开始，我们首先证明了在材料重量非常一般的假设下，横向电磁麦克斯韦算子的色散面上存在狄拉克点。我们对材料重量的假设几乎是确保一般六方光子晶体中狄拉克点存在的最低要求。然后我们在蜂窝结构被弱调制的情况下推导出相关的波包动力学。事实证明，缩减包络方程通常是一个二维非线性狄拉克方程，其质量随空间变化。通过研究具有类似畴壁的质量项的简化包络方程，我们实现了手性且不受局部缺陷影响的微妙波动。底层机制是存在拓扑保护的线性线模式，也称为边缘状态。然而，我们表明这些鲁棒的线性模式不具有非线性。我们证明了非线性线模式的存在，它可以基于高精度数值计算在非线性介质中传播。此外，我们还报告了一种在两个方向都局部化的新型非线性模式。