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Publications (218)
The paper focuses on the KPP–Fisher equation (named after Andrey Kolmogorov, Ivan Petrovskii, Nikolai Piskunov and Ronald Fisher) with non-local competitive losses and fractal time derivative which is considered in terms of Fα-calculus on the Cantor set dimension 0 < α < 1. A dynamic system with the fractal time derivative relating to the moments n...
We deal with the $n$-dimensional nonlinear Schr\"{o}dinger equation (NLSE) with a cubic nonlocal nonlinearity and an anti-Hermitian term, which is widely used model for the study of open quantum system. We construct asymptotic solutions to the Cauchy problem for such equation within the formalism of semiclassical approximation based on the Maslov c...
The present note is a feedback on the article by M.O. Katanaev in Physica Scripta (2023, 98, 104 001), where, in our opinion, a distorted view of the classical theory of separation of variables in the Hamilton-Jacobi equation is given. We show that the metrics given in this paper, addmiting separation of variables, are special cases of V.N. Shapova...
We construct quasiparticles-like solutions to the one-dimensional Fisher-Kolmogorov-Petrovskii-Piskunov (FKPP) with a nonlocal nonlinearity using the method of semiclassically concentrated states in the weak diffusion approximation. Such solutions are of use for predicting the dynamics of population patterns using analytical or semi-analytical appr...
The nonlinear Schrödinger equation (NLSE) with a non-Hermitian term is the model for various phenomena in nonlinear open quantum systems. We deal with the Cauchy problem for the nonlocal generalization of multidimensional NLSE with a non-Hermitian term. Using the ideas of the Maslov method, we propose the method of constructing asymptotic solutions...
The aim of this study was to apply integrative physiological mathematical models to simulate physiological parameters in traumatic shock caused by lower limb blast injury.
Materials and methods. At the first stage of mathematical modeling, we applied lumped parameter integrative physiological models, and at the second stage we used neural networks....
On the example of a linear one-dimensional heat equation and the related Burgers equation, symmetries in space with an additional independent variable that is not included in the equation, termed (parametrically) extended, are considered. Such symmetries are constructed using the extension of the non-commutative symmetry subalgebras of the equation...
We study the leading term of asymptotics for a solution to the two-dimensional kinetic equation with a nonlocal cubic nonlinearity that is a model of the ionization of active medium. The asymptotic solution under consideration is constructed analytically in the class of trajectory concentrated function within the approximation of weak diffusion. Th...
We construct quasiparticles-like solutions to the one-dimensional Fisher-Kolmogorov-Petrovskii-Piskunov (FKPP) with a nonlocal nonlinearity using the method of semiclassically concentrated states in the weak diffusion approximation. Such solutions are of use for predicting the dynamics of population patterns. The interaction of quasiparticles stems...
The nonlinear Sch\"{o}dinger equation (NLSE) with a non-Hermitian term is the model for various phenomena in nonlinear open quantum systems. We deal with the Cauchy problem for the nonlocal generalization of multidimensional NLSE with a non-Hermitian term. Using the ideas of the Maslov method, we propose the method of constructing asymptotic soluti...
Discrete kinetic equations describing binary processes of agglomeration and fragmentation are considered using formal equivalence between the kinetic equations and the geodesic equations of some affinely connected space A associated with the kinetic equation and called the kinetic space of affine connection. The geometric properties of equations ar...
The one-parameter two-dimensional cellular automaton with the Margolus neighbourhood is analyzed based on considering the projection of the stochastic movements of a single particle. Introducing the auxiliary random variable associated with the direction of the movement, we reduce the problem under consideration to the study of a two-dimensional Ma...
We study the known coherent states of a quantum harmonic oscillator from the standpoint of the originally developed noncommutative integration method for linear partial differential equations. The application of the method is based on the symmetry properties of the Schrödinger equation and on the orbit geometry of the coadjoint representation of Li...
We study the known coherent states of a quantum harmonic oscillator from the standpoint of the original developed noncommutative integration method for linear partial differential equations. The application of the method is based on the symmetry properties of the Schr\"odinger equation and on the orbit geometry of the coadjoint representation of Li...
Nonlocal versions of the reaction-diffusion type population equations can describe the evolution of spatiotemporal structures (patterns) depending on the equation parameter domain. Under conditions of weak diffusion, numerical methods have been used to compare the processes of spatiotemporal pattern formation in a nonlocal population model describe...
We propose a new approach that allows one to reduce nonlinear equations on Lie groups to equations with a fewer number of independent variables for finding particular solutions of the nonlinear equations. The main idea is to apply the method of noncommutative integration to the linear part of a nonlinear equation, which allows one to find bases in...
The one-parameter two-dimensional cellular automaton with the Margolus neighbourhood is analyzed based on the considering the projection of the stochastic movements of a single particle. Introducing the auxiliary random variable associated with the direction of the movement, we reduce the problem under consideration to the study of a two-dimensiona...
We apply the original semiclassical approach to the kinetic ionization equation with the nonlocal cubic nonlinearity in order to construct the family of its asymptotic solutions. The approach proposed relies on an auxiliary dynamical system of moments of the desired solution to the kinetic equation and the associated linear partial differential equ...
We apply the original semiclassical approach to the kinetic ionization equation with the nonlocal cubic nonlinearity in order to construct the family of its asymptotic solutions. The approach proposed relies on an auxiliary dynamical system of moments of the desired solution to the kinetic equation and the associated linear partial differential equ...
The general construction of the Cauchy problem solution for the one-dimensional nonlocal population Fisher–KPP equation is briefly described in terms of semiclassical asymptotics based on the complex WKB-Maslov method. For the particular case of the equation under consideration, a family of leading terms of the semiclassical asymptotics is construc...
A semiclassical approach based on the WKB–Maslov method is developed for the kinetic ionization equation in dense plasma with approximations characteristic of metal vapor active media excited by a contracted discharge. We develop the technique for constructing the leading term of the semiclassical asymptotics of the Cauchy problem solution for the...
A semiclassical approach based on the WKB-Maslov method is developed for the kinetic ionization equation in dense plasma with approximations characteristic of metal vapor active media excited by a contracted discharge. We develop the technique for constructing the leading term of the semiclassical asymptotics of the Cauchy problem solution for the...
Based on the group analysis of differential equations, we consider the symmetry properties of equations with fractal derivatives defined within the framework of Fα-calculus. Analogs of the prolongation of transformations of independent and dependent variables are discussed. The infinitesimal invariance of equations with fractal derivatives is studi...
We propose a new approach that allows one to reduce nonlinear equations on Lie groups to equations with a fewer number of independent variables for finding particular solutions of the nonlinear equations. The main idea is to apply the method of noncommutative integration to the linear part of a nonlinear equation, which allows one to find bases in...
We propose the approach to constructing semiclassical spectral series for the generalized multidimensional stationary Gross–Pitaevskii equation with a nonlocal interaction term. The eigenvalues and eigenfunctions semiclassically concentrated on a curve are obtained. The curve is described by the dynamic system of moments of solutions to the nonloca...
We develop a non-commutative integration method for the Dirac equation in homogeneous spaces. The Dirac equation with an invariant metric is shown to be equivalent to a system of equations on a Lie group of transformations of a homogeneous space. This allows us to effectively apply the non-commutative integration method of linear partial differenti...
We develop a noncommutative integration method for the Dirac equation in homogeneous spaces. The Dirac equation with an invariant metric is shown to be equivalent to a system of equations on a Lie group of transformations of a homogeneous space. This allows us to effectively apply the noncommutative integration method of linear partial differential...
We propose an approach to constructing semiclassical solutions for the generalized multidimensional Gross–Pitaevskii equation with a nonlocal interaction term. The key property of the solutions is that they are concentrated on a one-dimensional manifold (curve) that evolves over time. The approach reduces the Cauchy problem for the nonlocal Gross–P...
We consider an approach to constructing approximate analytical solutions for the one-dimensional twocomponent reaction-diffusion model describing the dynamics of population interacting with the active substance surrounding the population. The system of model equations includes the nonlocal generalized Fisher–Kolmogorov–Petrovsky–Piskunov equation f...
The Adomian decomposition method is applied to construct an approximate solution of the generalized one-dimensional Fisher–Kolmogorov–Petrovsky–Piskunov equation describing the population dynamics with nonlocal competitive losses. An approximate solution is constructed in the class of decreasing functions. The diffusion operator is taken as a rever...
We consider the effects of vacuum polarization and particle creation of a scalar field on Lie groups with a non-stationary bi-invariant metric of the Robertson–Walker type. The vacuum expectation values of the energy momentum tensor for a scalar field determined by the group representation are found using the noncommutative integration method for t...
We consider the effects of vacuum polarization and particle creation of a scalar field on Lie groups with a non-stationary bi-invariant metric of the Robertson-Walker type. The vacuum expectation values of the energy momentum tensor for a scalar field determined by the group representation are found using the noncommutative integration method for t...
We propose an approximate analytical approach to a ( 1 + 1 ) dimensional two-component system consisting of a nonlocal generalization of the well-known Fisher–Kolmogorov–Petrovskii– Piskunov (KPP) population equation and a diffusion equation for the density of the active substance solution surrounding the population. Both equations of the system ha...
A self-consistent model of the dynamics of a cellular population described by the generalized Fisher–Kolmogorov–Petrovskii–Piskunov equation with nonlocal competitive losses and interaction with the environment is formulated, in which the dynamics is described by the diffusion equation with allowance for the interaction of the population and the en...
The ability to diagnose oral lichen planus (OLP) based on saliva analysis using THz time-domain spectroscopy and chemometrics is discussed. The study involved 30 patients (2 male and 28 female) with OLP. This group consisted of two subgroups with the erosive form of OLP (n = 15) and with the reticular and papular forms of OLP (n = 15). The control...
The Fokker–Planck equation in one-dimensional spacetime with quadratically nonlinear nonlocal drift in the quasilocal approximation is reduced with the help of scaling of the coordinates and time to a partial differential equation with a third derivative in the spatial variable. Determining equations for the symmetries of the reduced equation are d...
This review deals with ideas and approaches to nonlinear phenomena, based on different branches of physics and related to biological systems, that focus on how small impacts can significantly change the state of the system at large spatial scales. This problem is very extensive, and it cannot be fully resolved in this paper. Instead, some selected...
A semiclassical approximation approach based on the Maslov complex germ method is considered in detail for the one-dimensional nonlocal Fisher–Kolmogorov–Petrovskii–Piskunov (Fisher–KPP) equation under the supposition of weak diffusion. In terms of the semiclassical formalism developed, the original nonlinear equation is reduced to an associated li...
The modified Fisher–Kolmogorov–Petrovskii–Piskunov equation with quasilocal quadratic competitive losses and variable coefficients in the small nonlocality parameter approximation is reduced to an equation with a nonlinear diffusion coefficient. Within the framework of a perturbation method, equations are obtained for the first terms of an asymptot...
We obtain and analyze equations determining first-order differential symmetry operators with matrix coefficients for the Dirac equation with an external electromagnetic potential in a $(2+1)$-dimensional Riemann (curved) spacetime. Nonequivalent complete sets of mutually commuting symmetry operators are classified in a $(2+1)$-dimensional Minkowski...
We obtain and analyze equations determining first-order differential symmetry operators with matrix coefficients for the Dirac equation with an external electromagnetic potential in a $(2+1)$-dimensional Riemann (curved) spacetime. Nonequivalent complete sets of mutually commuting symmetry operators are classified in a $(2+1)$-dimensional Minkowski...
The one-dimensional Fokker–Planck–Kolmogorov equation with a special type of nonlocal quadratic nonlinearity is represented as a consistent system of differential equations, including a dynamical system describing the evolution of the moments of the unknown function. Lie symmetries are found for the consistent system using methods of classical grou...
Noncommutative integration of the Klein–Gordon and Dirac relativistic wave equations in (2+1)-dimensional
Minkowski space is considered. It is shown that for all non-Abelian subalgebras of the (2+1)-dimensional
Poincaré algebra the condition of noncommutative integrability is satisfied.
A quasistationary solution of a two-component system of first-order telegraph equations on an interval with time-dependent conditions is constructed, where these conditions are prescribed at interior points of the interval. Application of the obtained solution as a criterion for leakage detection is considered.
The problem of thermal interaction between the gold nanoparticle heated by pulse-periodic laser radiation and the biotissue in which the nanoparticle is placed is numerically solved. A linear dependence between the heating time of the medium and the ratio of the time between laser radiation pulses to the time of the pulse for particles equally spac...
We examined possibilities of the Kalman filter for reducing the noise effects in the analysis of absorption spectra of gas samples, in particular, for samples of the exhaled air.
It has been shown that when comparing groups of patients with broncho-pulmonary diseases on the basis of the absorption spectra analysis of exhaled air samples the data pr...
In this paper we construct asymptotic solutions for the nonlocal multidimensional Fisher–Kolmogorov–Petrovskii–Piskunov equation in the class of functions concentrated on a one-dimensional manifold (curve) using a semiclassical approximation technique. We show that the construction of these solutions can be reduced to solving a similar problem for...
We consider an integro-differential 2-component multidimensional Gross-Pitaevskii equation with a Manakov-type cubic nonlocal nonlinearity. In the framework of the WKB-Maslov semiclassical formalism, we obtain a semiclassically reduced 2-component nonlocal Gross- Pitaevskii equation determining the leading term of the semiclassical asymptotic solut...
We explore the problem of thermal interaction of nanoparticles heated by laser radiation with a biological tissue after particle flow entering the cell. The solution of the model equations is obtained numerically under the following assumptions: a single particle is located in a neighborhood exceeding the particle size; the environm ent surrounding...
The results of numerical simulation of application principal component analysis to absorption spectra of breath air of patients with pulmonary diseases are presented. Various methods of experimental data preprocessing are analyzed. Keywords: breath air, pulmonary diseases, principal component analysis, data preprocessing 1. INTRODUCTION The directi...
We consider the problem of finding concentrations of molecular gases in the models of exhaled air samples in terms of their absorption spectra. We introduce model spectra describing the exhaled air samples as linear combinations of the absorption spectra of individual molecular gases with given coefficients. The absorption spectra are calculated on...
An approach to the reduction of the space of the absorption spectra, based on the original criterion for profile analysis of the spectra, was proposed. This criterion dates back to the known statistics chi-square test of Pearson. Introduced criterion allows to quantify the differences of spectral curves.
The possibility of the inverse spectroscopic problem solution for multicomponent gas mixtures based on the use of principal component analysis is discussed. The analysis revealed usefulness of principal component analysis to estimate the parameters of the components in a case of investigation of exhaled air samples from various groups of patients.
The results of comparison of quality of two classificators – SVM (support vector machine) and SIMCA (soft independent modelling of class analogies) on model data contained profiles of absorbtion specra of exhalted air are presented. It is shown, that SVM classification results can be improved by preprocessing if input data with principal component...
Canonical correlation analysis is adapted to the problem of determining the concentration of molecular components contained in samples of exhaled air. To solve this problem dealt with model spectra in form of linear combination of the absorption spectra of molecular components with unknown coefficients. The absorption spectra were calculated on the...
We explore methods of signal filtering using solutions of diffusion-type nonlinear and nonlocal model
equations as filter kernels. Basic feature of the considered filtering is replacement of commonly used Gaussian filter by a filter based on solutions of diffusion-type equations.
Within the formalism of the Fokker–Planck equation, the influence of nonstationary external force, random force, and dissipation effects on dynamics local conformational perturbations (kink) propagating along the DNA molecule is investigated. Such waves have an important role in the regulation of important biological processes in living systems at...
The comparison results of different mother wavelets used for de-noising of model and experimental data which were presented by profiles of absorption spectra of exhaled air are presented. The impact of wavelets de-noising on classification quality made by principal component analysis are also discussed.
Asymptotic solutions of the multidimensional nonlocal Fisher–Kolmogorov–Petrovskii–Piskunov equation with an influence function that is invariant with respect to a spatial shift are constructed. The asymptotic solutions are perturbations of a spatially-homogeneous quasistationary exact solution. General expressions are illustrated by the example of...
Integration of the Dirac equation with an external electromagnetic field is
explored in the framework of the method of separation of variables and of the
method of noncommutative integration. We have found a new type of solutions
that are not obtained by separation of variables for several external
electromagnetic fields. We have considered an exam...
Semiclassical asymptotics of the two-dimensional nonlocal Gross–Pitaevskii equation are constructed. The dynamics of the initial state, being a superposition of two wave packets, is investigated. The discrepancy of the obtained solution is investigated. The constructed asymptotic solutions are interpreted as a description of the interaction of two...
We have analyzed the data on concentration of geochemical elements within the 3- DV regional geochemical field. We have determined the fractal dimension index of spider diagrams and found the fractal nature of the distribution of chemical elements along the profile. We have compared these data with the results of the correlation analysis and hierar...
We consider vacuum polarization of a scalar field on the Lie groups with a bi-invariant metric
of Robertson-Walker type. Using the method of orbits we found expression for the vacuum expectation values
of the energy-momentum tensor of the scalar field which are determined by the representation character of the
group. It is shown that Einstein’s equ...
Asymptotic solutions of the nonlocal, one-dimensional Fisher–Kolmogorov–Petrovskii–Piskunov equation with fractional derivatives in the diffusion operator are constructed. The fractional derivative is defined in accordance with the approaches of Weyl, Grünwald–Letnilkov, and Liouville. Asymptotic solutions are constructed in a class of functions th...
With the help of the method of similarity transformations, an approach is considered that makes it possible to find particular solutions of the Gross–Pitaevskii equation with a nonstationary coefficient of nonlinearity in prolate spheroidal coordinates. Two exact solutions are found in explicit form, having soliton properties, along with the corres...
We consider the Dirac equation with external Yang-Mills gauge field in a
homogeneous space with invariant metric. The Yang-Mills fields for which the
motion group of the space serves as symmetry group of the Dirac equation are
found by comparison of the Dirac equation with a invariant matrix differential
operator of the first order. General constru...
The Dirac operator with an external Yang–Mills gauge field is considered on de Sitter space in terms of a noncommutative integration method related to the orbit method in the Lie group theory. A Yang–Mills field is presented for which the de Sitter group serves as the symmetry group of the Dirac operator. A spectrum of the Dirac operator with the Y...
Two analytical methods have been developed for constructing approximate
solutions to a nonlocal generalization of the 1D
Fisher-Kolmogorov-Petrovskii-Piskunov equation. This equation is of special
interest in studying the pattern formation in microbiological populations.
In the greater part of the paper, we consider in detail a semiclassical
approx...
A class of nonlinear symmetry operators has been constructed for the many-dimensional nonlocal Fisher-Kolmogorov-Petrovskii-Piskunov equation quadratic in independent variables and derivatives. The construction of each symmetry operator includes an interwining operator for the auxiliary linear equations and additional nonlinear algebraic conditions...
We have investigated the pattern formation in systems described by the
nonlocal Fisher--Kolmogorov--Petrovskii--Piskunov equation for the cases where
the dimension of the pattern concentration area is less than that of
independent variables space. We have obtained a system of integro-differential
equations which describe the dynamics of the concent...
Проводится сравнительный анализ методов (на основе показателей Липшица, Ляпунова, критерия Forward – Backward) оценки качества работы алгоритмов слежения (Mean-Shift и Particle filter) для использования в задачах видеонаблюдения. Исследуются потери объекта в видеопотоке, обусловленные перекрытием объекта слежения другими объектами или препятствиями...
The discrepancy of semiclassical asymptotics for the one-dimensional nonlocal Fisher–Kolmogorov–Petrovskii–Piskunov equation is investigated. It is shown that there exist values of the parameters of the system, for which the norm of the discrepancy is bounded and the accuracy of the asymptotic solution is preserved over the entire time interval, bu...
We consider symmetry properties of an integro-differential multidimensional
Gross-Pitaevskii equation with nonlocal cubic nonlinearity in the context of
symmetry analysis using the formalism of semiclassical asymptotics. This yields
a semiclassically reduced nonlocal Gross--Pitaevskii equation which determines
the principal term of a semiclassical...
Within the formalism of the Fokker-Planck equation, the influence of nonstationaryregular external force, random force, and dissipation on the kink dynamics is investigatedin the sine-Gordon model. The evolution equation of the kink momentum is written inthe form of the stochastic differential equation in the Stratonovich sense within theframework...
The classical group analysis approach used to study the symmetries of integro-differential equations in a semiclassical approximation is considered for a class of nearly linear integro-differential equations. In a semiclassical approximation, an original integro-differential equation leads to a finite consistent system of differential equations who...
Peculiarities of propagation of short optical pulses through resonantly absorbing fractal media are investigated in the work using methods of numerical simulation. The fractal object was modeled by a one-dimensional structure with its parameters being similar to generalized Cantor dust. The dependence of transmission of such a medium on its Hausdor...
Numerical solutions of the generalized one-dimensional Fisher–Kolmogorov–Petrovskii–Piskunov equation with nonlocal competitive
losses and convection are constructed. The influence function for nonlocal losses is chosen in the form of a Gaussian distribution.
The effect of convection on the dynamics of the spatially inhomogeneous distribution of th...
Solutions of a generalized Fisher–Kolmogorov–Petrovskii–Piskunov equation for a nonlocal interaction of finite radius have
been constructed for initial conditions with one and two localization centers by using numerical methods. The dynamics depends
on the choice of the equation parameters and initial conditions. The processes of formation and inte...
The two-dimensional Kolmogorov–Petrovskii–Piskunov–Fisher equation with nonlocal nonlinearity and axially symmetric coefficients
in polar coordinates is considered. The method of separation of variables in polar coordinates and the nonlinear superposition
principle proposed by the authors are used to construct the asymptotic solution of a Cauchy pr...
In the framework of the Lagrangian formalism the partial differential equation under study does not define univocally the Lagrangian density. In this paper we obtain a necessary and sufficient consistency condition over the ansatz that assures the invariance of the collective coordinates (CCs) equations under the change of equivalent Lagrangian den...
An analytical expression of the function modulating the amplitude of soliton velocity perturbations for the sine-Gordon equation
with a uniform external harmonic force and dissipation is derived. The evolution of the soliton velocity for the external
force in the form of a step function of time is examined. An analytical expression for the time-av...
A formalism of semiclassical asymptotics has been developed for a two-component Hartree-type evolutionary equation with a
small asymptotic parameter multiplying the partial derivatives, a nonlocal cubic nonlinearity, and a Hermite matrix operator.
Semiclassical solutions are constructed in the class of two-component functions concentrated in the ne...
A model of the evolution of a bacterium population based on the Fisher–Kolmogorov equation is considered. For a one-dimensional
equation of the Fisher–Kolmogorov type that contains quadratically nonlinear nonlocal kinetics and weak diffusion terms, a
general scheme of semiclassically concentrated asymptotic solutions is developed based on the compl...
The Fokker-Planck equation with diffusion coefficient quadratic in space variable, linear drift coefficient, and nonlocal nonlinearity term is considered in the framework of a model of analysis of asset returns at financial markets. For special cases of such a Fokker-Planck equation we describe a construction of exact solution of the Cauchy problem...
The Fokker-Planck equation with diffusion coefficient quadratic in space variable, linear drift coefficient, and nonlocal nonlinearity term is considered in the framework of a model of analysis of asset returns at financial markets. For special cases of such a Fokker-Planck equation we describe a construction of exact solution of the Cauchy problem...
Within the formalism of the Fokker-Planck equation, the influence of nonstationary external force, random force, and dissipation
effects on the kink dynamics is investigated in the sine-Gordon model. The equation of evolution of the kink momentum is obtained
in the form of the stochastic differential equation in the Stratonovich sense within the fr...
With the help of energy analysis suggested by McLaughlin and Scott for the sine-Gordon equation, evolution of kink velocity
modeling the propagation of a local conformational perturbation along the DNA molecule under the simultaneous action of dissipation
effects and special nonstationary external fields is investigated. For a harmonically time-dep...
We explore in detail the creation of stable localized structures in the form of localized energy distributions that arise from general initial conditions in the Peyrard-Bishop (PB) model. By means of a method based on the inverse scattering transform we study the solutions of PB model equations obtained in the form of planar waves whose amplitudes...
The Cauchy problem for the Gross--Pitaevsky equation with quadratic nonlocal nonlinearity is reduced to a similar problem for the correspondent linear equation. The relation between symmetry operators of the linear and nonlinear Gross--Pitaevsky equations is considered.
