On the existence of infinitely many nonperturbative solutions in a transmission eigenvalue problem for nonlinear Helmholtz equation with polynomial nonlinearity

•Phenomenon of nonlinear electromagnetic wave propagation is considered.•The problem is formulated with physically realistic conditions.•It is proved the existence of a novel (nonlinear) guided regime.•The existence of a nonperturbative effect is proved.•An original analytic approach is used to stud...

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Bibliographic Details
Published in:Applied Mathematical Modelling Vol. 53; pp. 296 - 309
Main Author: Valovik, D.V.
Format: Journal Article
Language:English
Published: New York Elsevier Inc 01.01.2018
Elsevier BV
Subjects:
Asymptotic methods
Comparative analysis
Cubic-quintic nonlinearity
Eigenvalues
Eigenvectors
Electromagnetic radiation
Helmholtz equations
Maxwell’s equations
Nonlinear comparison theory
Nonlinear eigenvalue problem
Nonlinear systems
Nonlinearity
Periodic variations
Plane dielectric waveguide
Polynomial nonlinearity
Polynomials
Propagation
Wave propagation
Polynomial nonlinearity
Nonlinear comparison theory
Plane dielectric waveguide
Cubic-quintic nonlinearity
Nonlinear eigenvalue problem
Maxwell’s equations
ISSN:0307-904X, 1088-8691, 0307-904X
Online Access:Get full text
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Summary:•Phenomenon of nonlinear electromagnetic wave propagation is considered.•The problem is formulated with physically realistic conditions.•It is proved the existence of a novel (nonlinear) guided regime.•The existence of a nonperturbative effect is proved.•An original analytic approach is used to study the problem. The paper focuses on a transmission eigenvalue problem for nonlinear Helmholtz equation with polynomial nonlinearity which describes the propagation of transverse electric waves along a dielectric layer filled with nonlinear medium. It is proved that even if the nonlinearity coefficients are small, the nonlinear problem has infinitely many nonperturbative solutions, whereas the corresponding linear problem always has a finite number of solutions. This results in the theoretical existence of a novel type of nonlinear guided waves that exist only in nonlinear guided systems. Asymptotic distribution of the eigenvalues is found and a comparison theorem is proved; periodicity of the eigenfunctions is proved, the exact formula for the period is found, and the zeros of the eigenfunctions are determined. The results found essentially extend the theory evolved earlier (particular cases for Kerr, cubic-quintic, septic nonlinearities, etc. are easily extracted from the general results found here). Numerical results are also presented.
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ISSN:0307-904X
1088-8691
0307-904X
DOI:10.1016/j.apm.2017.09.019