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Siberian Mathematical Journal

, Volume 60, Issue 4, pp 661–672 | Cite as

Plane Wave Solutions to the Equations of Electrodynamics in an Anisotropic Medium

  • V. G. RomanovEmail author
Article
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Abstract

Under examination is the system of equations of electrodynamics for a nonconducting nonmagnetic medium with the simplest anisotropy of permittivity. We assume that permittivity is characterized by the diagonal matrix ϵ = diag(ε1, ε1, ε2), with the functions ε1 and ε2 equal to positive constants beyond a bounded convex domain Ω ⊂ ℝ3. Two modes of traveling plane waves exist in a homogeneous anisotropic medium. The structure is studied of the solutions related to the traveling plane waves incident from infinity on an inhomogeneity located in Ω.

Keywords

Maxwell’s equations anisotropy plane wave structure of a solution 

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References

  1. 1.
    Romanov V. G., “Structure of a fundamental solution to the Cauchy problem for Maxwell’s equations,” Differ. Uravn., vol. 22, no. 9, 1577–1587 (1986).MathSciNetGoogle Scholar
  2. 2.
    Romanov V. G., Inverse Problems of Mathematical Physics, VNU Science Press, Utrecht, The Netherlands (1987).Google Scholar
  3. 3.
    Romanov V. G., “The problem of recovering the permittivity coefficient from the modulus of the scattered electromagnetic field,” Sib. Math. J., vol. 58, no. 4, 711–717 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Romanov V. G., “Problem of determining the permittivity in the stationary system of Maxwell equations,” Dokl. Math., vol. 95, no. 3, 230–234 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chadan K. and Sabatier P. C., Inverse Problems in Quantum Scattering Theory, Springer-Verlag, New York (1977).CrossRefzbMATHGoogle Scholar
  6. 6.
    Klibanov M. V. and Romanov V. G., “The first solution of a long standing problem: Reconstruction formula for a 3-d phaseless inverse scattering problem for the Schrödinger equation,” J. Inverse Ill-Posed Probl., vol. 23, no. 4, 415–428 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Klibanov M. V. and Romanov V. G., “Explicit solution of 3-D phaseless inverse scattering problems for the Schrödinger equation: the plane wave case,” Euras. J. Math. Comput. Appl., vol. 3, no. 1, 48–63 (2015).Google Scholar
  8. 8.
    Novikov R. G., “Explicit formulas and global uniqueness for phaseless inverse scattering in multidimensions,” J. Geom. Anal., 2015. DOI:  https://doi.org/10.1007/5.12220-014-9553-7.
  9. 9.
    Novikov R. G., “Formulas for phase recovering from phaseless scattering data at fixed frequency,” Bull. Sci. Math., 2015. DOI:  https://doi.org/10.1016/j.bulsci.2015.04.005.
  10. 10.
    Klibanov M. V. and Romanov V. G., “Explicit formula for the solution of the phaseless inverse scattering problem of imaging of nanostructures,” J. Inverse Ill-Posed Probl., vol. 23, no. 2, 187–193 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Romanov V. G., “Some geometric aspects in inverse problems,” Eurasian J. Math. Comput. Appl., vol. 3, no. 4, 68–84 (2015).Google Scholar
  12. 12.
    Klibanov M. V. and Romanov V. G., “Reconstruction procedures for two inverse scattering problem without the phase information,” SIAM J. Appl. Math., vol. 76, no. 1, 178–196 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Klibanov M. V. and Romanov V. G., “Two reconstruction procedures for a 3D phaseless inverse scattering problem for the generalized Helmholtz equation,” Inverse Problems, vol. 32, no. 2, 015005 (16 pp.) (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Klibanov M. V. and Romanov V. G., “Uniqueness of a 3-D coefficient inverse scattering problem without the phase information,” Inverse Problems, vol. 33, no. 9. 095007 (10 pp.) (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Romanov V. G. and Yamamoto M., “Phaseless inverse problems with interference waves,” J. Inverse Ill-Posed Probl., vol. 26, no. 5, 681–688 (2018).MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Romanov V. G. and Yamamoto M., “Recovering two coefficients in an elliptic equation via phaseless information,” Inverse Probl. Imaging, vol. 13, no. 1, 81–91 (2019).MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Romanov V. G., “Phaseless inverse problems that use wave interference,” Sib. Math. J., vol. 59, no. 3, 494–505 (2018).MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Courant R., Partial Differential Equations [Russian translation], Mir, Moscow (1964).zbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Inc. 2019

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirskRussia

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