Explicit determination is considered of the spectrum of normal waves in an inhomogeneous plane-parallel dielectric layer with a parabolic permittivity profile. Exact formulas are obtained for the propagation constants and normal wave field components.

A method for the accurate calculation of the cut-off wavenumbers of a waveguide with an arbitrary cross section and a number of inner conductors is demonstrated. A secure basis for formulating the spectral problem relies upon concepts of integral and infinite-matrix (summation) operator-valued functions depending nonlinearly on the frequency spectral parameter; whilst a variety of methods might be employed in determining the cut-off wavenumbers so specified, the Method of Analytical Regularization provides a route to the construction of an algorithm that is well-conditioned (and hence reliable). The algorithm is based on a mathematically rigorous solution of the homogeneous Dirichlet problem for the Helmholtz equation in the interior of the waveguide, excluding the regions occupied by the inner conductor boundaries; it results in a highly efficient method of calculating the cut-off wavenumbers and the corresponding non-trivial solutions representing the modal distribution. The ability to calculate the cut-off wavenumbers with any prescribed and proven accuracy provides a secure basis for treating the wavenumbers so found as "benchmark solutions". 

A method for the accurate calculation of the cut-off wavenumbers of a waveguide with an arbitrary cross section and a number of inner conductors is demonstrated. Concepts of integral and infinite-matrix (summation) operator-valued functions depending nonlinearly on the frequency spectral parameter provide a secure basis for formulating the spectral problem, and the Method of Analytical Regularization is employed to implement an effective algorithm. The algorithm is based on a mathematically rigorous solution of the homogeneous Dirichlet problem for the Helmholtz equation in the interior of the waveguide, excluding the regions occupied by the inner conductor boundaries. A highly efficient method of calculating the cut-off wavenumbers and the corresponding non-trivial solutions representing the modal distribution is developed. The mathematical correctness of the problem statement, the method, and the ability to calculate the cut-off wavenumbers with any prescribed and proven accuracy provide a secure basis for treating these as “benchmark solutions”. In this paper, we use this new approach to validate previously obtained results against our benchmark solutions. Furthermore, we demonstrate its universality in solving some new problems, which are barely accessible by existing methods.

The paper is devoted to the analysis of the TE-wave propagation in a planar waveguide with variable permittivity. The setting is reduced to a transmission eigenvalue problem for an ordinary differential equation where the spectral parameter is the wave propagation constant. For the computation of eigenvalues the shooting method is applied in the form specifically developed for this class of problems. The method allows one to calculate propagating, evanescent, and complex surface and leaky waves. A variety of numerical results illustrating the method is presented.

A brief survey is given of the most valuable contributions of Victor P. Shestopalov to electromagnetics and diffraction theory related to the application of the spectral theory of operator-valued functions. A particular attention is paid to the creation of the theory of intertype interaction of oscillations and waves in open structures on the basis of analysis of critical points of abstract dispersion equations. Contributions to the theory made by other researchers are reported.

The problem of scattering of electromagnetic waves on a homogeneous dielectric cylinder with a circular cross section is a classical task. Using the Fourier method it is possible to find an explicit solution written in the form of series [1]–​[3]. A rigorous justification of the Fourier method for solving this problem was first given in [4], [5].

Analysis of the scattering of electromagnetic waves by various open rectangular and cylindrical resonators and waveguides aimed at discovery of resonance effects has been carried out by many authors. However, an exhaustive answer and justification of the true mathematical nature of resonance effects in the general case has not been given until recent years. For a rectangular metal waveguide, mathematical justification of the existence of complex resonance frequencies is performed in [9]. Note also the works [6]–​[8] in which resonance phenomena in rectangular waveguides with metal inhomogeneities are studied using the method of partial regions. A rigorous proof of the existence of complex resonance frequencies associated with the plane wave scattering problem by a circular uniform dielectric cylinder is presented in [10], [11].

The present work is a continuation of [11]. The problem of diffraction of an E-polarized electromagnetic wave on a circular dielectric cylinder is studied with an objective to investigate the resonance scattering modes. A semi-analytical model is developed. Explicit form of the scattered-field expansion coefficient available for open cylindrical metal-dielectric scatterers with layered piece-wise homogeneous dielectric filling and circular symmetry enables one to correctly formulate the problem of finding the desired resonance parameters in terms of implicit complex-valued functions of one or several complex variables. Using the explicitly determined reference parameter values at which several expansion coefficients are simultaneously singular the Cauchy problem is formulated and solved within the model framework for the numerical determination of the spectral parameters corresponding to double and triple multiple-resonance scattering modes.

Non-stationary flooding of oil-saturated reservoirs has a long-standing and durable place as the main secondary method of oil production and maintenance of reservoir pressure in the development of most oil reservoirs. The water injection into the reservoir creates a delayed problem - the inevitable, often catastrophic flooding of oil production wells, provoked by a sudden and irreversible change in water saturation. The theory of two-phase flow filtration created by Buckley and Leverett does not take into account the loss of stability of the displacement front, which provokes an abrupt change and a triplicity of the water saturation value. Therefore, a mathematically simplified approach was proposed at one time, a repeatedly differentiable approximation to exclude a “jump” in water saturation. Such a simplified solution led to well-known negative consequences of the waterflooding practice, which experts call the “viscous instability of the displacement front”, the “fingering displacement front”. This work has presented a novel approach to formulation decisive rules for the first time allowing timely detection and prevention of the consequences of loss of stability of the displacement front and targeted control of the flooding system by stopping, forcing, limiting operating modes, assigning workover solutions of producing and injection wells. It is possible to quickly solve important short-term practical tasks passing traditional labor- intensive incorrect deterministic tasks and complex methods of solution mobilizing the injected water and controlling the fluid production rate, more precisely water and oil on the basis of the discriminant criterion.

We propose and develop a novel rigorous technique that enables one to obtain the explicit numerical values of parameters at which several lowest-order harmonics of the scattered field are suppressed. This provides partial cloaking of the object, a perfectly conducting cylinder of circular cross section covered by two layers of dielectric separated by an infinitely thin impedance layer, a two-layer impedance Goubau line (GL). The developed approach is a rigorous method that enables one to obtain in the closed form (and without numerical calculations) the values of parameters providing the cloaking effect, achieved particularly in terms of the suppression of several scattered field harmonics and variation of the sheet impedance. This issue constitutes the novelty of the accomplished study. The elaborated technique could be applied to validate the results obtained by commercial solvers with virtually no limitations on the parameter ranges, i.e., use it as a benchmark. The determination of the cloaking parameters is straightforward and does not require computations. We perform comprehensive visualization and analysis of the achieved partial cloaking. The developed parameter-continuation technique enables one to increase the number of the suppressed scattered-field harmonics by appropriate choice of the impedance. The method can be extended to any dielectric-layered impedance structures possessing circular or planar symmetry. 

We consider eigenvalue problems for dielectric cylindrical scatterers of arbitrary cross section with generalized conditions at infinity that enable one to take into account complex eigenvalues. The existence of resonance (scattering) frequencies associated with these eigenvalues is proved. The technique involves the determination of characteristic numbers (CNs) of the Fredholm operator-valued functions of the problems constructed using Green's potentials. Separating principal parts in the form of meromorphic operator pencils, we apply the operator generalization of Rouché's theorem to verify the occurrence of CNs in close proximities of the pencil poles. The results are illustrated in detail using the case of a dielectric cylinder of circular cross section.

The notion and instrument of dispersion curves (DCs) and generalized DCs are widely used [1] as a form to establish a relation between a quantity (e.g. eigen- or natural frequencies) that governs resonances or normal waves (often it is a spectral parameter of the corresponding problem) or cloaking (when the suppression of several scattered-field harmonics takes place) and a non-spectral (complex) variable, e.g. permittivity. Generally, reliance of the propagation, resonance and cloaking phenomena on various quantities characterizing the scatterers and the incident field, e.g. the frequency dependence of the propagation constants of normal waves that gives rise to DCs, is considered in terms of the parameter dependence of one or several quantities on one or several parameters. In this respect, such parameter dependences are naturally connected with certain mappings (establishing relations between domains or manifolds) rather than with individual (families of) functions of one variable, as in the case of DCs. The present work formulates the prerequisites of this approach generalizing the DC method, taking the case when such mappings are specified as complex functions of a complex variable, and considering a homogeneous dielectric cylinder of circular cross-section as a reasonable basic example.

The approaches and ideas exploring the application of conformal mappings to the analysis of cloaking and invisibility proposed in [2] and surveyed in [3] have been used within a different ‘coordinate’ framework. One can find in [3] and the works cited therein several examples of mappings that can be useful also when the present method of investigating scattered-field expansion coefficients is applied.