The paper focuses on the problem of normal surface waves in an open metal-dielectric inhomogeneous waveguide of arbitrary cross-section with losses. The medium filling the waveguide is characterized by a isotropic inhomogeneous complex permittivity. The setting is reduced to a boundary eigenvalue problem for longitudinal components of the electromagnetic field in Sobolev spaces. Variational formulation of the problem is considered in terms of the analysis of operator-functions. The discreteness of the set of the sought-for eigenvalues is proved. 

The behavior and properties of solutions of two-dimensional quadratic polynomial dynamical system on the phase plane of variables and time are considered. A complete qualitative theory is constructed which includes the analysis of all singular points and the features of solutions depending on all parameters of the problem. A main result is that, with the discriminant criteria created on the basis of the growth model constructed in the present study, it is possible to formulate practical recommendations for regulating and monitoring the process of waterflooding and the development of an oil field.

Statements and analysis are presented of nonselfadjoint eigenvalue problems for ellipticequations and systems, including singular Sturm–Liouville problems on the line, that arise inmathematical models of the wave propagation in open metal-dielectric waveguides. Existence ofreal and complex spectra are proved and their distribution is investigated for canonical structurespossessing circular symmetry of boundary contours.

The problem of propagation of electromagnetic waves in a inhomogeneous dielectric layer coated on one side with a layer of graphene, which is considered to be infinitely thin, is considered. The main problem in describing the process of wave propagation in a waveguide structure is to obtain a properties of the propagation constants. Maxwell’s equations are solved in the frequency domain. The conjugation conditions contain the conductivity of graphene. In this work, we neglect the nonlinearity of graphene. The results of calculations of propagation constants depending on the parameters of the problem are presented. 

The problem on leaky waves in an anisotropic inhomogeneous dielectric waveguide 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.