Mathematical Theory of Normal Waves in an Anisotropic Rod
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Mathematical Theory of Normal Waves in an Anisotropic Rod Yu. Smirnov1, 2* , E. Smolkin1, 2** , and M. Snegur1, 2*** (Submitted by E. K. Lipachev) 1
2
Penza State University, Penza, 440026 Russia Sirius University of Science and Technology, Sochi, 354340 Russia Received March 6, 2020; revised March 15, 2020; accepted March 20, 2020
Abstract—The problem on normal waves in an anisotropic inhomogeneous dielectric rod is considered. This problem is reduced to the boundary eigenvalue problem for longitudinal components of electromagnetic field in Sobolev spaces. To find the solution, we use the variational formulation of the problem. The variational problem is reduced to study of an operator-function. Discreteness of the spectrum is proved and distribution of the characteristic numbers of the operator-function on the complex plane is found. DOI: 10.1134/S1995080220070392 Keywords and phrases: non-polarized electromagnetic waves, inhomogeneous anisotropic waveguide, dielectric rod, non-linear eigenvalue problem, Maxwell’s equations, operatorfunction, spectrum.
1. INTRODUCTION Analysis of wave propagation in open waveguides constitutes an important class of vector electromagnetic problems. An anisotropic inhomogeneous dielectric rod is studied in the paper. The typical here are non self-adjoint boundary eigenvalue problems for the systems of Helmholtz equations with piecewise constant coefficients and transmission conditions containing the spectral parameter. On the discontinuity lines (surfaces) additional conditions called transmission conditions should be satisfied. However, the spectral parameter appears only usually in the equations and does not enter into transmission conditions, and so we have an eigenvalue problem for the usually self-adjoint operator. Sometimes, however, the spectral parameter occurs not only in the equation, but also in the transmission conditions, and often nonlinearly. We obtain a non self-adjoint problem. An approach based on the reduction to eigenvalue problems for operator pencils considered in Sobolev spaces was proposed by Smirnov in [1–3] (see also [4, 5]). General theory of polynomial operator-functions called operator pencils is sufficiently well elaborated. A fundamental work by Keldysh [6] pioneered investigation of non-self-adjoint polynomial pencils. The method of operator pencils is known to be a natural and efficient approach for investigation of the wave propagation in regular waveguides. Operator pencils were applied to the analysis of electromagnetic problems in [7–9]. Open waveguide structures were investigated by a number of authors [7, 10, 12]. However, for open (unshielded) structures, a complete theory of wave propagation is not constructed. In this case the problem becomes much more complicated (due to the non-compactness of the corresponding operators). The article deals with open anisotropic structures i.e. the case of an unbounded exterior domain is considered. The first results on the investigation of such problems were recently obtained in [13, 14] for the polarized waves p
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