Multiscale Modeling &Simulation, Volume 19, Issue 2, Page 951-979, January 2021.
In this article, we study wave dynamics in the fractional nonlinear Schrödinger equation with a modulated honeycomb potential. This problem arises from recent research interests in the interplay between topological materials and nonlocal governing equations. We first develop the Floquet--Bloch spectral theory of the linear fractional Schrödinger operator with a honeycomb potential. Especially, we prove the existence of conically 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.

中文翻译：

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具有蜂窝势的分数阶非线性薛定谔方程中的波包

多尺度建模与仿真，第 19 卷，第 2 期，第 951-979 页，2021 年 1 月。
在本文中，我们研究了具有调制蜂窝电势的分数阶非线性薛定谔方程中的波动力学。这个问题源于最近对拓扑材料和非局部控制方程之间相互作用的研究兴趣。我们首先开发了具有蜂窝势的线性分数薛定谔算子的 Floquet--Bloch 谱理论。特别地，我们证明了圆锥退化点的存在，即狄拉克点，两个色散带函数在该点相交。然后，我们研究了波包在狄拉克点光谱上的动力学，并推导出主要的有效包络方程。事实证明，包络线可以用质量变化的非线性狄拉克方程来描述。通过严格的误差估计，