Wave packet dynamics in slowly modulated photonic graphene

Abstract

Mathematical analysis on electromagnetic waves in photonic graphene, a photonic topological material which has a honeycomb structure, is one of the most important current research topics. By modulating the honeycomb structure, numerous topological phenomena have been observed recently. The electromagnetic waves in such a media are generally described by the 2-dimensional wave equation. It has been shown that the corresponding elliptic operator with a honeycomb material weight has Dirac points in its dispersion surfaces. In this paper, we study the time evolution of the wave packets spectrally concentrated at such Dirac points in a modulated honeycomb material weight. We prove that such wave packet dynamics is governed by the Dirac equation with a varying mass in a large but finite time. Our analysis provides mathematical insights to those topological phenomena in photonic graphene.

Wave packets in the fractional nonlinear Schrödinger equation with a honeycomb potential

“…Remark 4.4. This theorem gives the simplest form of asymptotic solution, which is parallel to that of linear wave packet problems [17,41]. However, one main difference is the lifetime of validity may not reach effective dynamics as (4.3) due to the case of nonlinearity.…”

“…|n| N , and it can be proved by the methods analogous to that used in [41]. Among the forthcoming derivation, we require N 4.…”

“…To deal with this nonlinear effect, we proceed to derive a second order approximation. The proof of Theorem 4.3 can be also followed by a modified spectral decomposition idea in the latter second order justification, which is referred from [17,41].…”

“…Now, we turn to the key estimate of G 2 (t, x). Although, the coefficient order here is lower than that of G 3 (t, x), we shall modify the procedure of subtle Bloch spectral decomposition presented in [17,41], and improve the estimate of G 2 (t, x) up to order of exactly ε 2 for t ∈ [0, T ].…”

“…Fefferman, Weinstein and their collaborators proved the existence of Dirac points lying on the dispersion surfaces of the honeycomb latticed standard Schödinger and divergence elliptic operators via Lyapunov-Schmidt reduction strategy [16,15,26,29]. Then, the time evolution of linear and nonlinear wave packet propagation spectrally concentrated around the Dirac point has been rigorously studied, and the effective dynamics is governed by corresponding Dirac equations [1,2,22,6,17,41]. Merging the two terminologies together in a semi-classical nonlinear evolution equation, we need to deal with mathematical challenges.…”