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Unified orbital description of the envelope dynamics in two-dimensional simple periodic lattices

M. J. Ablowitz University of Colorado, Boulder CO 80309 Yi Zhu Center for Applied Mathematics (ZCAM), Tsinghua University, Beijing 100084, China

TBD mathscidoc:2204.43005

2013.1
The propagation of wave envelopes in two-dimensional (2-D) simple periodic lattices is studied. A discrete approximation, known as the tight-binding (TB) approximation, is employed to find the equations governing a class of nonlinear discrete envelopes in simple 2-D periodic lattices. Instead of using Wannier function analysis, the orbital approximation of Bloch modes that has been widely used in the physical literature, is employed. With this approximation the Bloch envelope dynamics associated with both simple and degenerate bands are readily studied. The governing equations are found to be discrete nonlinear Schrödinger (NLS)-type equations or coupled NLS-type systems. The coefficients of the linear part of the equations are related to the linear dispersion relation. When the envelopes vary slowly, the continuous limit of the general discrete NLS equations are effective NLS equations in moving frames. These continuous NLS equations (from discrete to continuous) also agree with those derived via a direct multiscale expansion. Rectangular and triangular lattices are examples.
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@inproceedings{m.2013unified,
  title={Unified orbital description of the envelope dynamics in two-dimensional simple periodic lattices},
  author={M. J. Ablowitz, and Yi Zhu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220421112829183436081},
  year={2013},
}
M. J. Ablowitz, and Yi Zhu. Unified orbital description of the envelope dynamics in two-dimensional simple periodic lattices. 2013. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220421112829183436081.
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