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On the infinitely many nonperturbative solutions in a transmission eigenvalue problem for Maxwell’s equations with cubic nonlinearity

### Abstract

The paper focuses on a transmission eigenvalue problem for Maxwell’s equations with cubic nonlinearity that describes the propagation of transverse magnetic waves along the boundaries of a dielectric layer filled with nonlinear (Kerr) medium. Using an original approach, it is proved that even for small values of the nonlinearity coefficient, the nonlinear problem has infinitely many nonperturbative solutions (eigenvalues and eigenwaves), whereas the corresponding linear problem always has a finite number of solutions. This fact implies the theoretical existence of a novel type of eigenwaves that do not reduce to the linear ones in the limit in which the nonlinear coefficient reduces to zero. Asymptotic distribution of the eigenvalues is found, periodicity of the eigenfunctions is proved, the exact formula for the period is found, and the zeros of the eigenfunctions are determined.

Published by AIP Publishing.

Received 01 March 2016
Accepted 21 September 2016
Published online 07 October 2016

Acknowledgments:
The authors are supported by the Ministry of Education and Science of the Russian Federation (Goszadanie, Project No. 2.1102.2014K),and the second author is supported by the Russian Foundation for Basic Research (Project No. 15-01-00206) and the Russian Federation President Grant (Project No. MK-4684.2016.1).

Article outline:

I. INTRODUCTION
II. STATEMENT OF THE PROBLEM
III. LINEAR CASE
IV. MAIN RESULTS
V. PROOFS
A. First integral and determination of unknowns *X*_{0}, *X*_{h}, and B
B. Proof of theorem 1
C. Proof of theorem 2
D. Proof of theorem 3
VI. DISCUSSION AND CONCLUSION

/content/aip/journal/jmp/57/10/10.1063/1.4964279

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A. D. Boardman, P. Egan, F. Lederer, U. Langbein, and D. Mihalache, in Third-Order Nonlinear Electromagnetic TE and TM Guided Waves, edited by H.-E. Ponath and G. I. Stegeman (Elsevier Science Publishing, 1991), reprinted from Nonlinear Surface Electromagnetic Phenomena.

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D. V. Valovik and Yu. G. Smirnov, “Nonlinear effects in the problem of propagation of TM electromagnetic waves in a Kerr nonlinear layer,” J. Commun. Technol. Electron. 56(3), 283–288 (2011).

http://dx.doi.org/10.1134/S1064226911030120
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K. A. Yuskaeva, “On the theory of TM-electromagnetic guided waves in a nonlinear planar slab structure,” Ph.D. thesis, Universität Osnabrück, Universität Osnabrück Fachbereich Physik Barbarastraße 7, D - 49076 Osnabrück Germany, August 2012.

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Yu. G. Smirnov and D. V. Valovik, “Guided electromagnetic waves propagating in a plane dielectric waveguide with nonlinear permittivity,” Phys. Rev. A 91(1), 013840-1–013840-6 (2015).

http://dx.doi.org/10.1103/PhysRevA.91.013840
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H. W. Schürmann and V. S. Serov, “Comment on “Guided electromagnetic waves propagating in a plane dielectric waveguide with nonlinear permittivity”,” Phys. Rev. A 92(5), 057803(3pp.) (2015).

http://dx.doi.org/10.1103/PhysRevA.92.057803
11.

Yu. G. Smirnov and D. V. Valovik, “Reply to the comment on “Guided electromagnetic waves propagating in a plane dielectric waveguide with nonlinear permittivity”,” Phys. Rev. A 92(5), 057804(2pp.) (2015).

http://dx.doi.org/10.1103/physreva.92.057804
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D. V. Valovik, “On the problem of nonlinear coupled electromagnetic TE-TM wave propagation,” J. Math. Phys. 54(4), 042902(14pp.) (2013).

http://dx.doi.org/10.1063/1.4799275
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Yu. G. Smirnov and D. V. Valovik, “Problem of nonlinear coupled electromagnetic TE-TE wave propagation,” J. Math. Phys. 54(8), 083502(13pp.) (2013).

http://dx.doi.org/10.1063/1.4817388
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L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Course of Theoretical Physics, Electrodynamics of Continuous Media (Butterworth-Heinemann, Oxford, 1993), Vol. 8.

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N. N. Akhmediev and A. Ankevich, Solitons, Nonlinear Pulses and Beams (Chapman and Hall, London, 1997).

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M. J. Adams, An Introduction to Optical Waveguides (John Wiley & Sons, Chichester, New York, Brisbane, Toronto, 1981).

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L. A. Vainstein, Electromagnetic Waves (Radio i Svyaz, Moscow, 1988), p. 440 (in Russian).

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I. G. Petrovsky, Lectures on Ordinary Differential Equations (Moscow State University, Moscow, 1984) (in Russian).

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H.-G. Unger, Planar Optical Waveguides and Fibres (Clarendon Press, Oxford, 1977).

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Yu. G. Smirnov and D. V. Valovik, “Coupled electromagnetic TE-TM wave propagation in a cylindrical waveguide with Kerr nonlinearity,” J. Math. Phys. 54(4), 043506(22pp.) (2013).

http://dx.doi.org/10.1063/1.4799276
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E. Yu. Smol’kin and D. V. Valovik, “Guided electromagnetic waves propagating in a two-layer cylindrical dielectric waveguide with inhomogeneous nonlinear permittivity,” Adv. Math. Phys. 2015, 1–11.

http://dx.doi.org/10.1155/2015/614976
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2016-10-07

2016-10-24

### Abstract

The paper focuses on a transmission eigenvalue problem for Maxwell’s equations with cubic nonlinearity that describes the propagation of transverse magnetic waves along the boundaries of a dielectric layer filled with nonlinear (Kerr) medium. Using an original approach, it is proved that even for small values of the nonlinearity coefficient, the nonlinear problem has infinitely many nonperturbative solutions (eigenvalues and eigenwaves), whereas the corresponding linear problem always has a finite number of solutions. This fact implies the theoretical existence of a novel type of eigenwaves that do not reduce to the linear ones in the limit in which the nonlinear coefficient reduces to zero. Asymptotic distribution of the eigenvalues is found, periodicity of the eigenfunctions is proved, the exact formula for the period is found, and the zeros of the eigenfunctions are determined.

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