Traveling edge states in massive Dirac equations along slowly varying edges

Topologically protected wave motion has attracted considerable interest due to its novel properties and potential applications in many different fields. In this work, we study edge modes and traveling edge states via the linear Dirac equations with so-called domain wall 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 leading asymptotic solutions of two typical edge states that follow the circular and curved edges with small curvature by analytic and quantitative arguments.