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Wannier functions and discrete NLS equations for nematicons

  • Received: 12 November 2018 Accepted: 25 January 2019 Published: 12 March 2019
  • We derive nonlocal discrete nonlinear Schrödinger (DNLS) equations for laser beam propagation in optical waveguide arrays that use a nematic liquid crystal substrate. We start with an NLS-elliptic model for the problem and propose a simplified version that incorporates periodicity in one of the directions transverse to the propagation of the beam. We use Wannier basis functions for an associated Schrödinger operator with periodic potential to derive discrete equations for Wannier modes and propose some possible simplified systems for interactions of modes within the first energy band of the periodic Schrödinger operator. In particular, we present the simplest generalization of a model proposed by Fratalocchi and Assanto by including a linear nonlocal term, and see evidence for parameter regimes where nonlinearity is more pronounced.

    Citation: José Antonio Vélez-Pérez, Panayotis Panayotaros. Wannier functions and discrete NLS equations for nematicons[J]. Mathematics in Engineering, 2019, 1(2): 309-326. doi: 10.3934/mine.2019.2.309

    Related Papers:

  • We derive nonlocal discrete nonlinear Schrödinger (DNLS) equations for laser beam propagation in optical waveguide arrays that use a nematic liquid crystal substrate. We start with an NLS-elliptic model for the problem and propose a simplified version that incorporates periodicity in one of the directions transverse to the propagation of the beam. We use Wannier basis functions for an associated Schrödinger operator with periodic potential to derive discrete equations for Wannier modes and propose some possible simplified systems for interactions of modes within the first energy band of the periodic Schrödinger operator. In particular, we present the simplest generalization of a model proposed by Fratalocchi and Assanto by including a linear nonlocal term, and see evidence for parameter regimes where nonlinearity is more pronounced.


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