Method of Y-Mappings for Study of Multiparameter Nonlinear Eigenvalue Problems

For the study of nonlinear multiparameter eigenvalue problems, a method of Y -mappings, making it possible to prove the existence of solutions, is proposed. The problem of propagation of coupled polarized electromagnetic waves in a nonlinear layer with saturating nonlinearity is studied. The concept...

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Bibliographic Details
Published in:Computational mathematics and mathematical physics Vol. 62; no. 1; pp. 150 - 156
Main Author: Smirnov, Yu. G.
Format: Journal Article
Language:English
Published: Moscow Pleiades Publishing 01.01.2022
Springer Nature B.V
Subjects:
Computational Mathematics and Numerical Analysis
Eigenvalues
Eigenvectors
Electromagnetic radiation
Existence theorems
Fixed points (mathematics)
Mathematics
Mathematics and Statistics
Nonlinearity
Partial Differential Equations
Wave propagation
coupled polarized electromagnetic waves
multiparameter nonlinear eigenvalue problem
Sturm–Liouville problem
fixed point of mapping
ISSN:0965-5425, 1555-6662
Online Access:Get full text
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Summary:For the study of nonlinear multiparameter eigenvalue problems, a method of Y -mappings, making it possible to prove the existence of solutions, is proposed. The problem of propagation of coupled polarized electromagnetic waves in a nonlinear layer with saturating nonlinearity is studied. The concept of a Y -mapping, which puts into correspondence to the potential a special nonlinear function of several arguments: eigenfunctions of a linear problem, is defined. The multiparameter nonlinear eigenvalue problem is reduced to the problem of finding fixed points of Y -mappings. Using the Schauder theorem, the existence of an infinite set of fixed points of Y -mappings and, accordingly, solutions in a nonlinear multiparameter eigenvalue problem for sufficiently small values of the nonlinearity coefficient is proved.
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ISSN:0965-5425
1555-6662
DOI:10.1134/S0965542522010110