Semiclassical Propagation Along Curved Domain Walls

Multiscale Modeling &Simulation, Volume 22, Issue 1, Page 66-105, March 2024.
Abstract. We analyze the propagation of two-dimensional dispersive and relativistic wavepackets localized in the vicinity of the zero level set [math] of a slowly varying domain wall modeling the interface separating two insulating media. We propose a semiclassical oscillatory representation of the propagating wavepackets and provide an estimate of their accuracy in appropriate energy norms. We describe the propagation of relativistic modes along [math] and analyze dispersive modes by a stationary phase method. In the absence of turning points, we show that arbitrary smooth localized initial conditions may be represented as a superposition of such wavepackets. In the presence of turning points, the results apply only for sufficiently high-frequency wavepackets. The theory finds applications for both Dirac systems of equations modeling topologically nontrivial systems as well as Klein–Gordon equations, which are topologically trivial.