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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.


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 Montreal
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 Montreal | 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.


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.

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