V. M. Volkov

• Doctor of Physical and Mathematical Sciences
• Head of Department at Belarusian State University

. The issues of constructing numerical algorithms based on the Chebyshev spectral method for approximate solution of elliptic equations with mixed derivatives in a rectangular domain with homogeneous Dirichlet boundary conditions are considered. To implement the spectral method, the biconjugate gradients stabilized method with preconditioners in th...

We develop the theory of transportation and localization of a transparent dielectric spherical particle with the gradient forces in the interference field of orthogonally directed standing laser waves Ez(coskz) and Ex(coskx). It is shown that, when the waves Ez and Ex are coherent, the interference radiation field contains two harmonic components w...

We develop the theory of transportation and localization of a transparent dielectric spherical particle with the gradient forces in the interference field of orthogonally directed standing laser waves $E_z (\cos kz)$ and $E_x (\cos kx)$. It is shown that, when the waves $E_z$ and $E_x$ are coherent, the interference radiation field contains two har...

Development of efficient finite difference schemes and iterative methods for solving anisotropic diffusion problems in an arbitrary geometry domain is considered. To simplify the formulation of the Neumann boundary conditions, the method of fictitious domains is used. On the example of a two-dimensional model problem of potential distribution in an...

Рассмотрен спектральный метод Чебышева для двухточечных краевых задач, описывающих процессы встречного взаимодействия оптических волн в средах с кубической нелинейностью и линейных средах с периодической модуляцией показателя преломления. На примере линейной задачи показано, что для достижения заданной точности спектральный метод требует на два-три...

Finite difference schemes and iterative methods of solving anisotropic diffusion problems governing multidimensional elliptic PDE with mixed derivatives are considered. By the example of the test problem with discontinuous coefficients, it is shown that the spectral characteristics of the finite difference problem and the efficiency of their precon...

We investigate finite difference schemes and iterative methods for solving anisotropic diffusion problems governed by elliptic Partial Diffrential Equations (PDE) with mixed derivatives. On an example of a test problem with discontinuous coefficients, it is shown that the spectral characteristics of the finite difference problem and efficiency of t...

Advances in human brain neuroimaging for high-temporal and high-spatial resolution will depend on localization of Electroencephalography (EEG) signals to their cortex sources. The source localization inverse problem is inherently ill-posed and depends critically on the modeling of human head electromagnetics. We present a systematic methodology to...

Optimal parameters values minimizing the phase error of finite-difference schemes for non-stationary Schrodinger equation in the pre-defined spectral range are found. The obtained results are based on an equivalent representation of the θ-method in the form of family of one-parametric IIR-filters. Dependence of the optimal parameter value on the sp...

Spectral methods for a typical problem of counterpropagating optical wave interaction in nonlinear media are considered. Significant advantages of the spectral technique in compar-ison with 4-5-order spline-collocation methods are demonstrated. For implementation of non-linear spectral model a conservative iterative method is proposed. The proposed...

The electrical impedance tomortaphy (EIT) problems in anisotropic inhomogeneous media like head tissues belongs to the class of the three-dimensional boundary value problems for elliptic equations with mixed derivatives. The efficiency of the most discussed and usable in practice numerical methods in context of modeling EIT problems is reviewed in...

The Electrical Impedance Tomography (EIT) and electroencephalography (EEG) forward problems in anisotropic inhomogeneous media like the human head belongs to the class of the three-dimensional boundary value problems for elliptic equations with mixed derivatives. We introduce and explore the performance of several new promising numerical techniques...

Advances in human brain neuroimaging to achieve high-temporal and high-spatial resolution will depend on computational approaches to localize EEG signals to their sources in the cortex. The source localization inverse problem is inherently ill-posed and depends critically on the modeling of human head electromagnetics. In this paper we present a sy...

Based on numerical simulations, self-focusing of conventional and vortex optical beams produced by femtosecond pulses in air is comparatively analysed. It is shown that, other things being equal, in the case of self-focusing of vortex beams, a significantly higher concentration of energy is observed in the focal spot. As a consequence, there also a...

We considered the problems of computational modeling of oilfield geological structure and simulations of oil extraction process. The paper describes the mathematical model of non-steady displacement of non-Newtonian oil by water in essentially inhomogeneous bed tapped by a system of injecting and operating wells. It presents the algorithms of numer...

The problem of the generation of stable (quasi-stable) 3+1D light bullets in nonlinear media is addressed. We consider a medium with a two-component relaxing cubic nonlinearity, and non-relaxing quintic one. It is shown that a special adjustment of the pulse duration and the parameters of the two-component relaxing nonlinearity enables one not only...

Understanding the milliscale (temporal and spatial) dynamics of the human brain activity requires high-resolution modeling of head electromagnetics and source localization of EEG data. We have developed an automated environment to construct individualized computational head models from image segmentation and to estimate conductivity parameters usin...

Introduction. Inverse problems in neuroscience such as electroencephalography (EEG) and magnetoencephalograpy (MEG) source localization, electrical impedance tomography (EIT) or Magnetic Resonance EIT require high accuracy in measurements and PDE modeling. The emerging new reconstruction algorithms usually need a laboratory experiment on physical p...

We describe an efficient numerical method for solving the isotropic inhomogeneous 3D Poisson equation in cylindrical coordinates as applied to analysis of EIT phantom experimental systems. The proposed approach is based on the second order accuracy finite-difference scheme and the BiCG iterative method with the FFT preconditioner. Extensive validat...

We describe a novel 3D finite difference method for solving the anisotropic inhomogeneous Poisson equation based on a multi-component additive implicit method with a 13-point stencil. The serial performance is found to be comparable to the most efficient solvers from the family of preconditioned conjugate gradient (PCG) algorithms. The proposed mul...

The theory of formation of microspheres concentration structures in a suspension under the action of gradient forces in the field of counter-propagating laser beams is developed. The possibility of the microsphere pattern generation in suspensions with a mirror feedback is shown in diffusion limit on the basis of simultaneous solving of Planck-Nern...

A kinetic investigation was made of the operation of a distributed-feedback dye laser with a static phase grating in the travelling pump-wave regime. It was found that the use of a travelling pump wave makes it possible to implement unidirectional generation of a single ultrashort pulse of duration several times less than in the case of the traditi...

u (X1;x2;x3 )= u (x1;X2;x3 )= u (x1;x2;X3 )=0 : (2)

A multi-cluster computational environment with mixed-mode (MPI + OpenMP) parallelism for estimation of unknown regional electrical conductivities of the human head, based on realistic geometry from segmented MRI up to 2563 voxels resolution, is described. A finite difference multi-component alternating direction implicit (ADI) algorithm, paralleliz...

Based on the analysis of frequency-nondegenerate four-photon parametric scattering, the spectral-angular dependences of the increments of perturbing modes are obtained in the field of an intense light wave propagating in a medium with cubic nonlinearity.

Based on the analysis of frequency-nondegenerate four-photon parametric scattering, the spectral-angular dependences of the increments of perturbing modes are obtained in the field of an intense. light wave propagating in a medium with cubic nonlinearity.

The intrinsic optical bistability, soliton formation, and transient phenomena such as the free polarization decay and photon echo in dense resonant media are investigated, taking into account the short-range dipole-dipole interaction of atoms.

The optical soliton train propagation is studied under the action of intra pulse Raman scattering (IRS). It is shown that IRS induces decay of the soliton train at het early stage of this propagation. As a method to suppress the adverse action of IRS, a train propagation in a composite medium with fast and slow relaxing cubic non-linearities is ana...

Modified Bloch equations that describe the interaction of high-power optical radiation with a dense resonance medium consisting of two subsystems of different two-level atoms of high density are obtained. It is shown that, unlike the one-component resonance medium in which the effect of internal optical bistability manifests itself on the red wing...

The effect of relaxation of nonlinearity on the formation of a shock wave of the intensity envelope of ultrashort laser pulses is studied on the basis of numerical analysis of the corresponding truncated wave equation. The relaxation of nonlinearity is shown to lead to the stabilisation of the envelope sharpening process, with the minimum width of...

In the framework of the model of a two-component inertial cubic nonlinearity the dark soliton stability against the intrapulse Raman scattering is shown to be feasible under a certain relationship between parameters of a medium and pulse. Stability conditions are obtained for the pulse duration being between the times of fast and slow relaxations o...

The optical soliton train propagation is studied under the action of intra pulse Raman scattering (IRS). It is shown that IRS induces decay of the soliton train at het early stage of this propagation. As a method to suppress the adverse action of IRS, a train propagation in a composite medium with fast and slow relaxing cubic non-linearities is ana...

We investigate spatial instabilities (induced by small angular perturbations) of counterpropagating waves in nonlinear distributed feedback (DFB) structures. We determined the DFB-structure threshold length at which an absolute instability occurs and a nonhomogeneous spatial intensity distribution is generated. The evolution of the transverse inten...

The local-field effects are considered which are feasible in a dense ensemble of resonant atoms modeled by multilevel quantum systems. Our approach is based on the generalized two-level system [V.S. Butylkin, A.E. Kaplan, and Yu.G. Khronopulo, Zh. Éksp. Teor. Fiz. 59, 921 (1970)]. Making use of this model, we take account of the nonresonant polariz...

The response of a nonlinear Bragg grating under bidirectional illumination is studied. By analytical and numerical techniques it is found that the threshold for bistability, pulsation development, and spatial transverse instability is strongly influenced by the phase difference between fields incident from opposite directions. Breakup of pulsations...

A system of equations for pulse generation by a DFB dye laser with spatially modulated traveling- wave pumping is obtained and investigated using numerical simulation. Analytical expressions are derived for the coefficients of amplification and the coupling of counterpropagating pulses. It is shown that the traveling-wave pumping in a range of opti...

Kinetics of a DFB laser with a binary mixture of dyes (molecules of a donor and an acceptor) under travelling-waves conditions was examined. It is shown that nonlinear absorption of radiation by the molecules of the acceptor can lead to a considerable (several times) shortening of the duration of the output pulse.

The equation governing the propagation of spiral beams in a non-linear medium is analysed. It is shown that, taking into account the saturation effect, spiral beams have a tube-like structure with a periodicity along the axis of a non-linear autoguide. A critical power is found which it is necessary to exceed in order to give rise to the indicated...

Soliton propagation in a nonlinear cubic medium with two types of relaxing nonlinearities is investigated. Provided that the pulse duration exceeds the fast relaxation time and be shorter than the slow one, partial compensation of the concerted influence of these relaxations on the soliton dynamics is feasible. A condition for such an optimal regim...

We consider a boundary value problem of reflection of a powerful light wave from a dense resonant medium that exhibits intrinsic optical bistability because of the dipole–dipole interaction of two-level atoms. On the basis of analytical solutions and numerical simulations we establish that intrinsic optical bistability results in the dependence of...

A solution is obtained of the problem of the spatial structure of the output radiation field of a distributed-feedback laser with a static phase grating pumped longitudinally by a beam with a parabolic intensity profile. The spectrum of the normal transverse modes is found for a distributed-feedback laser. The spectrum is described by a superpositi...

The coupled-wave method is used to obtain the exact solution of the wave equations describing the propagation of optical radiation in an active distributed-feedback structure based on a phase grating with a linearly chirped period. The normal mode spectrum and the threshold gain are found for an active distributed-feedback structure with aperiodic...

V. I. Kruglov and V. M. Volkov Proceedings of IV Int. Sem. Nonlinear Phenomena in Complex Systems, 6-9 February, 1995, Minsk, Belarus.

The light transmission through a system of bistable thin films of two-level atoms (centres) is considered. In the limit of an inertialess two-level medium, the problem is reduced to a nonlinear infinite-dimension mapping. It is shown that self-pulsation regimes occur when the incident light intensity is not enough to switch all bistable elements in...

The light propagation through a system of coupled bistable thin films is considered. In the limit of short relaxation times the problem is reduced to a infinite-dimension map. It is shown that self-pulsation regimes occur when the incident light intensity is not high enough to switch all bistable elements in the upper state. In the case of the gaus...

We investigate the complex space-time propagation dynamics of soliton-like light pulses in a birefringent fiber. The dynamics can be described by means of a low-dimensional quasisolition attractor. The correlation dimension of this attractor is less than 3.

The analytical theory governing the propagation of spiral laser beams in nonlinear media is developed. The stability of a circular symmetry structure of the beam is investigated. It is found that critical autoguide power depends on the topological charge of the spiral beam.

The theory of light-induced phase conjugation in a cavity is developed for an FWM process with pump beams resulting from resonantly scattered emission the generation of which is due to nonstationary energy exchange. On the basis of a numerical solution of the equations describing this process we have investigated the generation kinetics of the refe...

The analytical theory governing the propagation of spiral laser beams in nonlinear media is developed. It is shown that, taking into account the saturation effect in a Kerr medium, spiral beams have a tube-like structure with a periodicity along the axis of a nonlinear autoguide. In the case of the absence of saturation mechanism the collapse of sp...

The hexagonal patterns in nonlinear optics previously have been discovered experimentally [1,2] and numerically [3]. It was noted that the appearance of the hexagons is one of the displaying of strong counter-propagated light beams instability, in particular, in the media with Kerr-like nonlinearity.

The interaction of counterpropagating self-induced transparency solitons is studied. The pulses' asymptotics after collision depending on the solitons' parameters before collision and the influence of diffusion ‘erasure’ of the population difference gratings on solitons' interaction are determined numerically.

The paper presents a theoretical study of the interaction of helical beams with arbitrary topological charges in the autowave propagation regime. The self-similar solution describing two-dimensional two-wave collapse is obtained. Finally, it is suggested that the feasibility of the experimental generation of such helical beams is connected with the...

The stationary effect of self-action of counterpropagating axially symmetric light beams in a medium with cubic nonlinearity is discussed. Self-similar solutions for initial radiation parameters and nonlinear medium characteristics in the case of slowly varying amplitudes of beams are found. The analysis of approximate solutions obtained on the bas...