On a Nonlinear Eigenvalue Problem Related to the Theory of Propagation of Electromagnetic Waves

The eigenvalue problem is studied for a quasilinear second-order ordinary differential equation on a closed interval with Dirichlet’s boundary conditions (the corresponding linear problem has an infinite number of negative and no positive eigenvalues). An additional (local) condition imposed at one...

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Bibliographic Details
Published inDifferential equations Vol. 54; no. 2; pp. 165 - 177
Main Author Valovik, D. V.
Format Journal Article
LanguageEnglish
Published Moscow Pleiades Publishing 01.02.2018
Springer
Springer Nature B.V
Subjects
Analysis
Difference and Functional Equations
Differential equations
Dirichlet problem
Eigenvalues
Eigenvectors
Electric waves
Electromagnetic radiation
Electromagnetic waves
Mathematics
Mathematics and Statistics
Ordinary Differential Equations
Partial Differential Equations
Perturbation theory
Wave propagation
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Summary:The eigenvalue problem is studied for a quasilinear second-order ordinary differential equation on a closed interval with Dirichlet’s boundary conditions (the corresponding linear problem has an infinite number of negative and no positive eigenvalues). An additional (local) condition imposed at one of the endpoints of the closed interval is used to determine discrete eigenvalues. The existence of an infinite number of (isolated) positive and negative eigenvalues is proved; their asymptotics is specified; a condition for the eigenfunctions to be periodic is established; the period is calculated; and an explicit formula for eigenfunction zeroes is provided. Several comparison theorems are obtained. It is shown that the nonlinear problem cannot be studied comprehensively with perturbation theory methods.
ISSN:0012-2661
1608-3083
DOI:10.1134/S0012266118020039