论文标题
具有蜂窝电位的分数非线性schrödinger方程中的波数据包
Wave packets in the fractional nonlinear Schrödinger equation with a honeycomb potential
论文作者
论文摘要
在本文中，我们研究了具有调制蜂窝电位的分数非线性schrödinger方程中的波动力学。这个问题来自拓扑材料与非本地管理方程之间的相互作用的最新研究兴趣。两者都是当前的专注于科学研究领域。我们首先开发了具有蜂窝电位的线性分数schrödinger操作员的浮标光谱理论。尤其是，我们证明存在圆锥变性点的存在，即dirac点，在该点上两个分散谱带函数相交。然后，我们研究了在dirac点上定位的波数据包的动力学，并得出了领先的有效包膜方程。事实证明，信封可以通过具有不同质量的非线性狄拉克方程来描述。通过严格的误差估计，我们证明了基于有效包膜方程的渐近解决方案近似于加权 - $ h^s $空间中的真实解决方案。
In this article, we study wave dynamics in the fractional nonlinear Schrödinger equation with a modulated honeycomb potential. This problem arises from recent research interests in the interplay between topological materials and nonlocal governing equations. Both are current focuses in scientific research fields. We first develop the Floquet-Bloch spectral theory of the linear fractional Schrödinger operator with a honeycomb potential. Especially, we prove the existence of conical degenerate points, i.e., Dirac points, at which two dispersion band functions intersect. We then investigate the dynamics of wave packets spectrally localized at a Dirac point and derive the leading effective envelope equation. It turns out the envelope can be described by a nonlinear Dirac equation with a varying mass. With rigorous error estimates, we demonstrate that the asymptotic solution based on the effective envelope equation approximates the true solution well in the weighted-$H^s$ space.
