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.

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Edge states attract more and more research interests owing to the novel topologically protected properties. In this work, we studied edge modes and traveling edge states via the linear Dirac equation with so-called edge-admissible masses. The unidirectional edge state provides a heuristic approach to more general traveling edge states through the localized behavior along slowly varying edges. We show the dominated asymptotic solutions of two typical edge states that follow circular and curved edges with small curvature by the analytic and quantitative arguments.

To understand the hidden physical concepts from observed data is the most basic but challenging problem in many fields. In this study, we propose a new type of interpretable neural network called the ordinary differential equation network (ODENet) to reveal the hidden dynamics buried in the massive time-series data. Specifically, we construct explicit models presented by ordinary differential equations (ODEs) to describe the observed data without any prior knowledge. In contrast to other previous neural networks which are black boxes for users, the ODENet in this work is an imitation of the difference scheme for ODEs, with each step computed by an ODE solver, and thus is completely understandable. Backpropagation algorithms are used to update the coefficients of a group of orthogonal basis functions, which specify the concrete form of ODEs, under the guidance of loss function with sparsity requirement. From classical Lotka-Volterra equations to chaotic Lorenz equations, the ODENet demonstrates its remarkable capability to deal with time-series data. In the end, we apply the ODENet to real actin aggregation data observed by experimentalists, and it shows an impressive performance as well.

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