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Open waveguides in doubly periodic junctions of domains with different limit dimensions

Abstract

Considering the spectral Neumann problem for the Laplace operator on a doubly periodic square grid of thin circular cylinders (of diameter ε ≪ 1) with nodes, which are sets of unit size, we show that by changing or removing one or several semi-infinite chains of nodes we can form additional spectral segments, the wave passage bands, in the essential spectrum of the original grid. The corresponding waveguide processes are localized in a neighborhood of the said chains, forming I-shaped, V-shaped, and L-shaped open waveguides. To derive the result, we use the asymptotic analysis of the eigenvalues of model problems on various periodicity cells.

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Correspondence to F. L. Bakharev.

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Original Russian Text Copyright © 2016 Bakharev F.L. and Nazarov S.A.

The authors were supported by St. Petersburg State University (Project 0.38.237.2014). The first author was also supported by the Chebyshev Laboratory of St. Petersburg State University, the Government of the Russian Federation (Agreement No. 11.G34.31.0026), and Gazprom Neft PJSC.

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Bakharev, F.L., Nazarov, S.A. Open waveguides in doubly periodic junctions of domains with different limit dimensions. Sib Math J 57, 943–956 (2016). https://doi.org/10.1134/S0037446616060021

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Keywords

  • spectral Neumann problem
  • doubly periodic grid
  • localized waves
  • open waveguides