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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Vydané v:	Applied Mathematical Modelling Ročník 53; s. 296 - 309
Hlavný autor:	Valovik, D.V.
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	New York Elsevier Inc 01.01.2018 Elsevier BV
Predmet:	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
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Popis
Shrnutí:	•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.
Bibliografia:	ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14
ISSN:	0307-904X 1088-8691 0307-904X
DOI:	10.1016/j.apm.2017.09.019