Wave packets in the fractional nonlinear Schr\"odinger equation with a honeycomb potential

TL;DR

This paper analyzes wave packet dynamics in the fractional nonlinear Schr"odinger equation with honeycomb potential, revealing Dirac points and deriving an effective nonlinear Dirac equation with rigorous error bounds.

Contribution

It develops Floquet-Bloch spectral theory for the fractional Schr"odinger operator with honeycomb potential and derives an effective nonlinear Dirac equation for wave packets near Dirac points.

Findings

Existence of Dirac points in the spectral band structure.

Derivation of an effective nonlinear Dirac equation with a varying mass.

Rigorous approximation of true solutions by asymptotic solutions in weighted-$H^s$ space.

Abstract

In this article, we study wave dynamics in the fractional nonlinear Schr\"odinger equation with a modulated honeycomb potential. This problem arises from recent research interests in the interplay between topological materials and nonlocal governing equations. Both are current focuses in scientific research fields. We first develop the Floquet-Bloch spectral theory of the linear fractional Schr\"odinger operator with a honeycomb potential. Especially, we prove the existence of conical degenerate points, i.e., Dirac points, at which two dispersion band functions intersect. We then investigate the dynamics of wave packets spectrally localized at a Dirac point and derive the leading effective envelope equation. It turns out the envelope can be described by a nonlinear Dirac equation with a varying mass. With rigorous error estimates, we demonstrate that the asymptotic solution based on the effective envelope equation approximates the true solution well in the weighted-$H^s$ space.