Том 59, № 4 (2019)

The asymptotics of the solution to the Helmholtz equation in a three-dimensional layer of variable thickness with a localized right-hand side in the absence of “trap” states and under the asymptotic radiation conditions at infinity is constructed. The wave part of the solution has a finite number of modes. The resulting formula makes sufficiently clear the influence of the shape of the source on the wave part of the solution.

A problem with data on the characteristics for a quasilinear hyperbolic equation is considered. An inverse problem consisting in determining the unknown coefficient of the equation, depending on its solution, is formulated. As additional information for the inverse problem, the values of the solution of the problem with the data on the characteristics at a fixed value of one of the independent variables are specified. The existence theorem for the inverse problem is proved. The proof is based on deriving a nonlinear operator equation for the unknown coefficient and the proof of its solvability.

A two-dimensional reaction–diffusion equation in a medium with discontinuous characteristics is considered; the existence, local uniqueness, and asymptotic stability of its stationary solution, which has a large gradient at the interface, is proved. This paper continues the authors’ works concerning the existence and stability of solutions with internal transition layers of boundary value problems with discontinuous terms to multidimensional problems. The proof of the existence and stability of a solution is based on the method of upper and lower solutions. The methods of analysis proposed in this paper can be generalized to equations of arbitrary dimension of the spatial variables, as well as to more complex problems, e.g., problems for systems of equations. The results of this work can be used to develop numerical algorithms for solving stiff problems with discontinuous coefficients.

Features of plane-parallel and axisymmetric flows of an ideal (inviscid and non-heat-conducting) gas with a straight sonic line are studied. The flows under study occur in symmetric and asymmetric plane nozzles and axisymmetric nozzles with circular and annular cross sections. The acceleration and deceleration of flows when a straight sonic line is approached or receded from are studied. Additionally, sonic streamlines are constructed, including a contour of a sonic central body, i.e., a sonic streamline starting at the point of a straight sonic line on the axis of symmetry. The capabilities of the developed approaches are illustrated by examples, which are of interest in themselves.

The problem of diffraction of a polarized electromagnetic wave by a layer filled with a nonlinear medium is considered. The layer is located between two half-spaces with constant permittivities. Two widely used types of nonlinearities: saturation nonlinearity and Kerr nonlinearity are considered. It is proved that the results on the solvability of the problems in these cases are qualitatively different: in the case of saturation nonlinearity, there are conditions under which the diffraction problem has a unique solution and, in the case of Kerr nonlinearity, the diffraction problem always has an infinite set of solutions. Analytical and numerical methods for solving this kind of problems are developed. Numerical results are presented.

An incompressible boundary layer on a compliant plate is considered. The influence exerted by the inertia of the plate on the stability of the boundary layer is studied in the limit of high Reynolds numbers on the basis of triple-deck theory. The flow is found to have two additional oscillation eigenmodes, one of which is always unstable, but grows more slowly than classical modes corresponding to Tollmien–Schlichting waves. It is shown that, with decreasing inertia of the plate, the perturbations first split into two wave packets, which later merge in a single one that grows progressively more quickly.