Traveling edge states in massive Dirac equations along slowly varying edges

Abstract

Abstract Topologically protected wave motion has attracted considerable research interest due to its chirality and potential applications in many applied fields. We construct quasi-traveling wave solutions to the 2D Dirac equation with a domain wall mass in this work. It is known that the system admits exact and explicit traveling wave solutions, which are termed edge states if the interface is a straight line. By modifying such explicit solutions, we construct quasi-traveling-wave solutions if the interface is nearly straight. The approximate solutions in two scenarios are given. One is the circular edge with a large radius, and the second is a straight line edge with the slowly varying along the perpendicular direction. We show the quasi-traveling wave solutions are valid in a long lifespan by energy estimates. Numerical simulations are provided to support our analysis both qualitatively and quantitatively.