Nonlinear Waves in Shallow Honeycomb Lattices

Mark J. Ablowitz Department of Applied Mathematics, University of Colorado, 526 UCB, Boulder, CO 80309-0526 Yi Zhu Zhou Pei-Yuan Center for Applied Mathematics, Tsinghua University, Beijing 100084, China

TBD mathscidoc:2204.43026

SIAM Journal on Applied Mathematics, 72, (1), 21, 2012.1
The linear spectrum and corresponding Bloch modes of shallow honeycomb lattices near Dirac points are investigated. Via perturbation theory, the dispersion relation is found to have threefold degeneracy at leading order with eigenvalue splitting at the following two orders; i.e., the threefold eigenvalue splits into single and double values. Multiscale perturbation methods are employed to describe the nonlinear dynamics of the associated wave envelopes. The dynamics of the envelope depends on different asymptotic balances whereupon a three-level nonlinear Dirac-type equation or a two-level nonlinear Dirac equation is derived. The analysis agrees well with direct numerical simulations.
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  title={Nonlinear Waves in Shallow Honeycomb Lattices},
  author={Mark J. Ablowitz, and Yi Zhu},
  booktitle={ SIAM Journal on Applied Mathematics},
Mark J. Ablowitz, and Yi Zhu. Nonlinear Waves in Shallow Honeycomb Lattices. 2012. Vol. 72. In SIAM Journal on Applied Mathematics. pp.21.
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