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Zh. Vychisl. Mat. Mat. Fiz., 2018, Volume 58, Number 10, Pages 1656–1665 (Mi zvmmf10792)  

This article is cited in 2 scientific papers (total in 2 papers)

On the existence of an infinite number of eigenvalues in one nonlinear problem of waveguide theory

D. V. Valovik, S. V. Tikhov

Penza State University, Penza, Russia

Abstract: A nonlinear Sturm–Liouville-type eigenvalue problem on an interval with a boundary condition of the first kind and an additional local condition at one of the boundaries of the interval is considered. All the parameters of the problem are real. The existence of an infinite number of (isolated) positive eigenvalues is proven, their asymptotic behavior is indicated, a condition for the periodicity of the eigenfunctions is found, the period is calculated, and an explicit formula for the zeros of the eigenfunction is presented. It is shown that methods of perturbation theory are not applicable to the complete study of the nonlinear problem.

Key words: nonlinear Sturm–Liouville-type eigenvalue problem, quasilinear differential equation, asymptotics of eigenvalues, comparison theorem.

Funding Agency Grant Number
Ministry of Education and Science of the Russian Federation 1.894.2017/4.6

DOI: https://doi.org/10.31857/S004446690003585-4

References: PDF file   HTML file

English version:
Computational Mathematics and Mathematical Physics, 2018, 58:10, 1600–1609

Bibliographic databases:

UDC: 517.984.5
Received: 25.10.2017
Revised: 23.01.2018

Citation: D. V. Valovik, S. V. Tikhov, “On the existence of an infinite number of eigenvalues in one nonlinear problem of waveguide theory”, Zh. Vychisl. Mat. Mat. Fiz., 58:10 (2018), 1656–1665; Comput. Math. Math. Phys., 58:10 (2018), 1600–1609

Citation in format AMSBIB
\by D.~V.~Valovik, S.~V.~Tikhov
\paper On the existence of an infinite number of eigenvalues in one nonlinear problem of waveguide theory
\jour Zh. Vychisl. Mat. Mat. Fiz.
\yr 2018
\vol 58
\issue 10
\pages 1656--1665
\jour Comput. Math. Math. Phys.
\yr 2018
\vol 58
\issue 10
\pages 1600--1609

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Tikhov V S., Valovik D.V., “Propagation of Electromagnetic Waves in a Shielded Dielectric Layer With Cubic Nonlinearity”, 2018 Days on Diffraction (Dd), eds. Motygin O., Kiselev A., Goray L., Kazakov A., Kirpichnikova A., Perel M., IEEE, 2018, 283–287  isi
    2. D. V. Valovik, S. V. Tikhov, “Linearizuemye i nelinearizuemye resheniya v nelineinoi zadache o sobstvennykh znacheniyakh, voznikayuschei v teorii elektrodinamicheskikh volnovodov, zapolnennykh nelineinoi sredoi”, MaterialyXVII Vserossiiskoi molodezhnoishkoly-konferentsii Lobachevskie chteniya-2018, 23-28 noyabrya 2018 g., Kazan.Chast 2, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 176, VINITI RAN, M., 2020, 34–49  mathnet  crossref
  •      Computational Mathematics and Mathematical Physics
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