Institute of Mathematics of NAS of Ukraine

Kiev, Ukraine

Institute of Mathematics of NAS of Ukraine

Kiev, Ukraine

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Listopadova V.,National Technical University of Ukraine | Magda O.,Kyiv National Economic University | Pobyzh V.,Institute of Mathematics of NAS of Ukraine
Nonlinear Analysis: Real World Applications | Year: 2013

It is shown that the forced Korteweg-de Vries (KdV) equation studied in the recent papers [A.H. Salas, Computing solutions to a forced KdV equation, Nonlinear Anal. RWA 12 (2011) 1314-1320] and [M.L. Gandarias, M.S. Bruzón, Some conservation laws for a forced KdV equation, Nonlinear Anal. RWA 13 (2012) 2692-2700] is reduced to the classical (constant-coefficient) KdV equation by point transformations for all values of variable coefficients. The equivalence-based approach proposed in [R.O. Popovych, O.O. Vaneeva, More common errors in finding exact solutions of nonlinear differential equations: part I, Commun. Nonlinear Sci. Numer. Simul. 15 (2010) 3887-3899] allows one to obtain more results in a much simpler way. © 2012 Published by Elsevier Ltd.


Vaneeva O.O.,Institute of Mathematics of NAS of Ukraine | O Popovych R.,Institute of Mathematics of NAS of Ukraine | O Popovych R.,Wolfgang Pauli Institute | Sophocleous C.,University of Cyprus
Physica Scripta | Year: 2014

We discuss how point transformations can be used for the study of integrability, in particular, for deriving classes of integrable variable-coefficient differential equations. The procedure of finding the equivalence groupoid of a class of differential equations is described and then specified for the case of evolution equations. A class of fifth-order variable-coefficient Korteweg-de Vries-like equations is studied within the framework suggested. © 2014 The Royal Swedish Academy of Sciences.


Gerasimenko V.I.,Institute of Mathematics of NAS of Ukraine | Tsvir Z.A.,Taras Shevchenko National University
Journal of Physics A: Mathematical and Theoretical | Year: 2010

We develop a rigorous formalism for describing the evolution of states of quantum many-particle systems in terms of a one-particle density operator. For initial states which are specified in terms of a one-particle density operator, the equivalence of the description of the evolution of quantum many-particle states by the Cauchy problem of the quantum Bogolyubov-Born-Green-Kirkwood-Yvon hierarchy, and by the Cauchy problem of the generalized quantum kinetic equation, together with a sequence of explicitly defined functionals of a solution of a stated kinetic equation, is established in the space of trace-class operators. The links between the specific quantum kinetic equations with the generalized quantum kinetic equation are discussed. © 2010 IOP Publishing Ltd.


Popovych R.O.,Institute of Mathematics of NAS of Ukraine | Popovych R.O.,Wolfgang Pauli Institute | Bihlo A.,University of Montréal
Journal of Mathematical Physics | Year: 2012

Methods for the design of physical parameterization schemes that possess certain invariance properties are discussed. These methods are based on different techniques of group classification and provide means to determine expressions for unclosed terms arising in the course of averaging of nonlinear differential equations. The demand that the averaged equation is invariant with respect to a subalgebra of the maximal Lie invariance algebra of the unaveraged equation leads to a problem of inverse group classification which is solved by the description of differential invariants of the selected subalgebra. Given no prescribed symmetry group, the direct group classification problem is relevant. Within this framework, the algebraic method or direct integration of determining equations for Lie symmetries can be applied. For cumbersome parameterizations, a preliminary group classification can be carried out. The methods presented are exemplified by parameterizing the eddy vorticity flux in the averaged vorticity equation. In particular, differential invariants of (infinite-dimensional) subalgebras of the maximal Lie invariance algebra of the unaveraged vorticity equation are computed. A hierarchy of normalized subclasses of generalized vorticity equations is constructed. Invariant parameterizations possessing minimal symmetry extensions are described and a restricted class of invariant parameterization is exhaustively classified. The physical importance of the parameterizations designed is discussed. © 2012 American Institute of Physics.


Bihlo A.,University of Montréal | Dos Santos Cardoso-Bihlo E.,University of Vienna | Popovych R.O.,Institute of Mathematics of NAS of Ukraine
Journal of Mathematical Physics | Year: 2012

Preliminary group classification became a prominent tool in the symmetry analysis of differential equations due to the paper by Ibragimov, Torrisi, and Valenti [J. Math. Phys.32, 2988-2995 (1991)10.1063/1.529042]. In this paper the partial preliminary group classification of a class of nonlinear wave equations was carried out via the classification of one-dimensional Lie symmetry extensions related to a fixed finite-dimensional subalgebra of the infinite-dimensional equivalence algebra of the class under consideration. In the present paper we implement the complete group classification of the same class up to both usual and general point equivalence using the algebraic method of group classification. This includes the complete preliminary group classification of the class and finding those Lie symmetry extensions which are not associated with subalgebras of the equivalence algebra. The complete preliminary group classification is based on listing all inequivalent subalgebras of the whole infinite-dimensional equivalence algebra whose projections are qualified as maximal extensions of the kernel algebra. The set of admissible point transformations of the class is exhaustively described in terms of the partition of the class into normalized subclasses. © 2012 American Institute of Physics.


Gerasimenko V.I.,Institute of Mathematics of NAS of Ukraine | Polishchuk D.O.,Taras Shevchenko National University
Mathematical Methods in the Applied Sciences | Year: 2011

We discuss the origin of the microscopic description of correlations in quantum many-particle systems obeying Fermi-Dirac and Bose-Einstein statistics. For correlation operators that give the alternative description of the quantum state evolution of Bose and Fermi particles, we deduce the von Neumann hierarchy of nonlinear equations and construct the solution of its initial-value problem in the corresponding spaces of sequences of trace class operators. The links of constructed solution both with the solution of the quantum BBGKY hierarchy and with the nonlinear BBGKY hierarchy for marginal correlation operators are discussed. The solutions of the Cauchy problems of these hierarchies are constructed, in particular for initial data satisfying a chaos property. Copyright © 2010 John Wiley & Sons, Ltd.


Borovchenkova M.S.,Taras Shevchenko National University | Gerasimenko V.I.,Institute of Mathematics of NAS of Ukraine
Journal of Physics A: Mathematical and Theoretical | Year: 2014

We develop a rigorous formalism for the description of the kinetic evolution of many-particle systems with dissipative interaction. The links of the evolution of a hard sphere system with inelastic collisions described within the framework of marginal observables governed by the dual Bogolyubov-Born- Green-Kirkwood-Yvon (BBGKY) hierarchy and the evolution of states described by the Cauchy problem of the Enskog kinetic equation for granular gases are established. Moreover, we consider the Boltzmann-Grad asymptotic behavior of the constructed non-Markovian Enskog kinetic equation for granular gases in a one-dimensional space. © 2014 IOP Publishing Ltd.


Pocheketa O.A.,Institute of Mathematics of NAS of Ukraine | Popovych R.O.,Institute of Mathematics of NAS of Ukraine | Popovych R.O.,Wolfgang Pauli Institute
Physics Letters, Section A: General, Atomic and Solid State Physics | Year: 2012

Reduction operators of generalized Burgers equations are studied. A connection between these equations and potential fast diffusion equations with power nonlinearity of degree -1 via reduction operators is established. Exact solutions of generalized Burgers equations are constructed using this connection and known solutions of the constant-coefficient potential fast diffusion equation. © 2012 Elsevier B.V.


Bihlo A.,University of Montréal | Popovych R.O.,Wolfgang Pauli Institute | Popovych R.O.,Institute of Mathematics of NAS of Ukraine
Physics Letters, Section A: General, Atomic and Solid State Physics | Year: 2012

Following our paper [A. Bihlo, R.O. Popovych, J. Math. Phys. 50 (2009) 123102 (12 pp.), arXiv:0902.4099], we systematically carry out Lie symmetry analysis for the barotropic vorticity equation on the rotating sphere. All finite-dimensional subalgebras of the corresponding maximal Lie invariance algebra, which is infinite-dimensional, are classified. Appropriate subalgebras are then used to exhaustively determine Lie reductions of the equation under consideration. The relevance of the constructed exact solutions for the description of real-world physical processes is discussed. It is shown that the results of the above paper are directly related to the results of the recent Letter [N.H. Ibragimov, R.N. Ibragimov, Phys. Lett. A 375 (2011) 3858] in which Lie symmetries and some exact solutions of the nonlinear Euler equations for an atmospheric layer in spherical geometry were determined. © 2012 Elsevier B.V. All rights reserved.


Nesterenko M.,Institute of Mathematics of NAS of Ukraine
Physics of Particles and Nuclei Letters | Year: 2014

Two main approaches to the construction of realizations are discussed. The practical calculation algorithm, based on the method of I. Shorokov is proposed. A new realization of the Poincaré algebra p(1, 3) is presented as an example. © 2014, Pleiades Publishing, Ltd.

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