On a nonlinear eigenvalue problem related to the theory of propagation of electromagnetic waves
type: Item
instance of: scholarly article
title: On a nonlinear eigenvalue problem related to the theory of propagation of electromagnetic waves
zbMATH Open document ID: 1453.34031
DOI: 10.1134/S0012266118020039
published in: Differential Equations
publication date: 2018-07-20T00:00:00Z
review text: The eigenvalue problem is studied for a quasilinear second-order ordinary differential equation on a closed interval with Dirichlets 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.
Mathematics Subject Classification ID: 34B09
Mathematics Subject Classification ID: 34L15
zbMATH DE Number: 6908987
zbMATH Keywords: nonlinear eigenvalue problem
zbMATH Keywords: discrete eigenvalues
zbMATH Keywords: infinite number of positive and negative eigenvalues
author: D. V. Valovik
reviewed by: Hanying Feng
MaRDI profile type: Publication
cites work: Integral dispersion equation method to solve a nonlinear boundary eigenvalue problem
cites work: Guided electromagnetic waves propagating in a two-layer cylindrical dielectric waveguide with inhomogeneous nonlinear permittivity
cites work: On the infinitely many nonperturbative solutions in a transmission eigenvalue problem for Maxwell's equations with cubic nonlinearity
cites work: Q4434430
cites work: Nonlinear double-layer Bragg waveguide: analytical and numerical approaches to investigate waveguiding problem
cites work: Q3234774
cites work: Dual variational methods in critical point theory and applications
cites work: Q3232510
cites work: Q4681938
cites work: On the existence of infinitely many eigenvalues in a nonlinear Sturm-Liouville problem arising in the theory of waveguides
cites work: Q3317362
full work available at URL: s0012266118020039
OpenAlex ID: W2799991370
Wikidata QID: Q129945737
description: scientific article; zbMATH DE number 6908987