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Nonlinear waves in shallow honeycomb lattices. (English) Zbl 1258.41012

Authors’ abstract: 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.

MSC:

41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
35C20 Asymptotic expansions of solutions to PDEs
35Q55 NLS equations (nonlinear Schrödinger equations)
35L60 First-order nonlinear hyperbolic equations
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