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

Peng Xie Center for Applied Mathematics (ZCAM), Tsinghua University, Beijing 100084, China Yi Zhu Yau Mathematical Science Center, Tsinghua University, Beijing, China

Analysis of PDEs mathscidoc:2204.03002

2020.6
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.
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@inproceedings{peng2020wave,
  title={Wave packets in the fractional nonlinear Schrödinger equation with a honeycomb potential},
  author={Peng Xie, and Yi Zhu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220422135846265938094},
  year={2020},
}
Peng Xie, and Yi Zhu. Wave packets in the fractional nonlinear Schrödinger equation with a honeycomb potential. 2020. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220422135846265938094.
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