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Dirichlet-to-Neumann Mapping for the Characteristic Elliptic Equations with Symmetric Periodic Coefficients

Published online by Cambridge University Press:  03 June 2015

Jingsu Kang*
Affiliation:
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, P.R. China
Meirong Zhang*
Affiliation:
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, P.R. China
Chunxiong Zheng*
Affiliation:
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, P.R. China
*
Corresponding author.Email:[email protected]
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Abstract

Based on the numerical evidences, an analytical expression of the Dirichlet-to-Neumann mapping in the form of infinite product was first conjectured for the one-dimensional characteristic Schrödinger equation with a sinusoidal potential in [Commun. Comput. Phys., 3(3): 641-658, 2008]. It was later extended for the general second-order characteristic elliptic equations with symmetric periodic coefficients in [J. Comp. Phys., 227: 6877-6894, 2008]. In this paper, we present a proof for this Dirichlet-to-Neumann mapping.

Type
Research Article
Copyright
Copyright © Global Science Press Limited 2014

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References

[1]Antoine, X., Arnold, A., Besse, C., Ehrhardt, M. and Schadle, A., A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrodinger equations, Commun. Comput. Phys., 4 (2008), 729796.Google Scholar
[2]Barth, M. and Benson, O., Manipulation of dielectric particles using photonic crystal cavities, Appl. Phys. Lett., 89 (2006), 253114.Google Scholar
[3]Bastard, G., Wave mechanics applied to semiconductor heterostructures, les editions de physique, Les Ulis Cedex, France, 1988.Google Scholar
[4]Ehrhardt, M. (ed.), Wave propagation in periodic media, Progress in Computational Physics, Vol. 1, Bentham Science Publishers Ltd., 2010.Google Scholar
[5]Ehrhardt, M. and Zheng, C., Exact artificial boundary conditions for problems with periodic structures, J. Comput. Phys., 227(2008), 68776894.Google Scholar
[6]Fliss, S. and Joly, P., Exact boundary conditions for time-harmonic wave propagation in locally perturbed periodic media, Appl. Numer. Math., 59 (2009), 21552178.Google Scholar
[7]Fox, C., Oleinik, V. and Pavlov, B., A Dirichlet-to-Neumann map approach to resonance gaps and bands of periodic networks, Contemp. Math., 412 (2006), 151170.CrossRefGoogle Scholar
[8]Givoli, D., Non-reflecting boundary conditions, J. Comput. Phys., 94 (1991), 129.Google Scholar
[9]Griffiths, D. J. and Steinke, C. A, Waves in locally periodic media, Am. J. Phys., 69 (2001), 137154.Google Scholar
[10]Hagstrom, T., Radiation boundary conditions for the numerical simulation of waves, Acta Numerica, 8 (1999), 47106.Google Scholar
[11]Han, Z., Forsberg, E. and He, S., Surface plasmon Bragg gratings formend in metal-insulator-metal waveguides, IEEE Photonics Techn. Lett., 19 (2007), 9193.Google Scholar
[12]Hoang, V., The limiting absorption principle for a periodic semi-infinite waveguide, SIAM J. Appl. Math., 71 (3) (2011), 791810.Google Scholar
[13]Hohage, T. and Soussi, S., Riesz bases and Jordan form of the translation operator in semi-infinite periodic waveguides, J. Math. Pures Appl., 100 (2013), 113135.Google Scholar
[14]Joly, P., Li, J.-R. and Fliss, S., Exact Boundary Conditions for Periodic Waveguides Containing a Local Perturbation, Commun. Comput. Phys., 1 (2006), 945973.Google Scholar
[15]Kuchment, P., The mathematics of photonic crystals, Mathematical Modeling in Optical Science, 22 (2001), 207272.Google Scholar
[16]Magnus, W. and Winkler, S., Hill’s Equation, Interscience Wiley, New York, 1979.Google Scholar
[17]Póschel, J. and Trubowitz, E., Inverse Spectral Theory, Academic Press, 1987.Google Scholar
[18]Sakoda, K., Optical Properties of Photonic Crystals, Springer-Verlag, Berlin, 2001.Google Scholar
[19]Sondergard, T., Bozhevolnyi, S. I. and Boltasseva, A., Theoretical analysis of ridge gratings for long-range surface plasmon polaritons, Phys. Rev. B, 73 (2006), 045320.Google Scholar
[20]Smith, D. R., Pendry, J. B. and Wiltshire, M. C. K., Metamaterials and negative refractive index, Science, 305 (2004), 788792.Google Scholar
[21]Tausch, J. and Butler, J., Floquet multipliers of periodic waveguides via Dirichlet-to-Neumann maps, J. Comput. Phys., 159 (2000), 90102.Google Scholar
[22]Tausch, J. and Butler, J., Efficient analysis of periodic dielectric waveguides using Dirichlet- to-Neumann maps, J. Opt. Soc. Amer. A, 19 (2002), 11201128.Google Scholar
[23]Tsynkov, S. V., Numerical solution of problems on unbounded domains, Appl. Numer. Math., 27 (1998), 465532.Google Scholar
[24]Wacker, A., Semiconductor superlattices: A model system for nonlinear transport, Phys. Rep., 357 (2002), 1111.Google Scholar
[25]Yuan, L. and Lu, Y. Y., Dirichlet-to-Neumann map method for second harmonic generation in piecewise uniform waveguides, J. Opt. Soc. of Am. B, 24 (2007), 22872293.Google Scholar
[26]Yuan, L. and Lu, Y. Y., A recursive doubling Dirichlet-to-Neumann map method for periodic waveguides, J. Lightwave Technology, 25 (2007), 36493656.Google Scholar
[27]Zhang, M., The rotation number approach to eigenvalues of the one-dimensional p-Laplacian with periodic potentials, J. London Math. Soc.-Second Ser., 64 (2001), 125143.Google Scholar
[28]Zheng, C., An exact absorbing boundary conditions for the Schrodinger equation with sinusoidal potentials at infinity, Commun. Comput. Phys., 3 (2008), 641658.Google Scholar