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We develop methods of construction of functionally invariant solutions U(x, y, z, t) for the nonlinear nonautonomic Klein-Fock-Gordon equation. The solutions U(x, y, z, t) are found in the form of an arbitrary function, that depends on one, τ(x, y, z, t), or two, α(x, y, z, t), β(x, y, z, t) specially constructed functions. The functions are called ansatzes. The ansatzes (τ,α,β) are defined as solutions...
We consider the control problem for the 2D model of the scattering TM-polarized electromagnetic waves. This problem arises when developing the design technologies of electromagnetic cloaking devices using optimization method. The role of control function is played by refraction index of the dielectric obstacle. We prove the solvability of the control problem, derive the optimality system, study the...
A nonlinear model of radiative-conductive heat transfer in a layered medium with Fresnel matching condition at the interfaces is considered. The P1 approximation of this model is derived. A numerical algorithm to solve the boundary-value problem is implemented. The comparison of models with and without accounting for reflection/refraction effects is fulfilled. The importance of taking into account...
We consider the dynamical inverse problem for two-velocity systems on finite trees in a time-optimal setting: i. e. we assume that the dynamical Dirichlet-to-Neumann map, which we use as inverse data, is known on some finite interval (the length of this interval depends on the optical diameter of a tree). Using the controllability of a dynamical system and ideas of the Boundary Control method, we...
We consider the beam operator on the unit interval. For such operator, we prove few trace formulas, similar to the Gelfand-Levitan formulas for second order operators.
Metasurfaces are artificial electromagnetic boundaries usually implemented as two-dimensional periodic arrays of engineered inclusions with sub-wavelength periodicity, which may serve as useful tools to control characteristics of electromagnetic fields. Recently the authors have proposed employing a class of frequency selective metasurfaces with self-complimentary patterns to construct optically thin...
A variant of “0D quantum field theory” alternative of random matrices is proposed. The Feynman's path integrals are directly replaced by usual multiple Riemannian ones over finite-dimensional real Euclidean space. In this scheme we realized L. D. Faddeev's version of background field formalism. As an example the ϕ4 model is discussed. Necessary Feynman diagram technics is constructed. If diagrams...
In this work, we study optical properties of all-dielectric metasurfaces on the top of high-index substrate. We perform numerical simulations of spherical nanoparticle arrays for different permittivities of the substrate, and propose a simple model with separate contributions from nanoparticle arrays and bare substrate into the total reflection, which fully agrees with numerical results. For the single...
Resonant tunneling is known to emerge in ballistic electron transport in a quantum waveguide with two narrows (constrictions). Here we present results of numerical simulations of electron transport in a system consisting of two identical resonators produced by three narrows in a quantum waveguide. It is shown that, at small electron energies, the resonance peaks of one-resonator system split and the...
Acoustoelasticity approach is a method of non-destructive testing and it is based upon the theory of propagation of ultrasonic longitudinal and transverse waves of different polarization in solids. This approach is able to uniquely determine the value of the principal stress in the case of no plastic deformation. An elastoplastic material with hardening is taken and the problem of propagation of a...
In this paper, experimental realization of the surface plasmon laser in a solution for the visible and near infrared region (VNIR) based on the hybrid nanoparticles with a 13-nanometre gold core or the nanorod with various aspect ratios surrounded by a 20–40 nm thick dye-embedded silica shell (fluorescein or DCM) was presented. The spasing in those structures using the methods of confocal microscopy,...
A Cauchy problem for the wave equation with data given on a time-like hyperplane is considered. The problem is ill-posed. Assuming the data to be such that the problem is solvable, we propose a constructive procedure, which provides the solution.
We consider the spectrum of the discrete Schrödinger equation with one-dimensional perturbation. We obtain an explicit form of scattering matrix and find an exact condition of the absence of the singular part of the spectrum. We also calculated the eigenvalue that appears if this condition is not true. In the last part of our paper, we give few remarks on the case of two-dimensional perturbations.
In this paper, we are concerned with the study of the spectrum of operators with periodic potentials in 1D and 2D, modelling the interactions of an electron with regular lattices of ions. The classical Bloch theory asserts that the spectrum has a band structure and we are able to localize the spectral bands which are in the range of the potential and to prove estimates on their widths in the semiclassical...
The steady-state regime of a moving load on an elastic half-plane is addressed. It is shown that the solution can be expressed through a single harmonic function, similarly to the known eigensolution for surface Rayleigh wave, thus reducing a vector problem in linear elasticity to a scalar one for the Laplace equation. Examples of steadily moving vertical force and punch are investigated, illustrating...
Developing the diffraction approach, we study the most interesting and also most complicated case of scattering problem of three three-dimensional unlike-charged quantum particles.
We consider the quantum scattering problem of three one-dimensional particles with repulsive shortrange pair potentials. For clarity, we restrict ourself by the case of finite pair potentials. The absence of singular continuous spectrum of the corresponding Schröedinger operator for a broad class of pair potentials was proved earlier in a known work of E. Mourre. Nevertheless, the Mourre techniques...
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