2018
Том 70
№ 9

All Issues

Prikarpatskii A. K.

Articles: 27
Article (Ukrainian)

On the completely integrable calogero-type discretizations of Lax-integrable nonlinear dynamical systems and related coadjoint Markov-type orbits

Prikarpatskii A. K.

↓ Abstract

Ukr. Mat. Zh. - 2016. - 68, № 5. - pp. 657-664

The Calogero-type matrix discretization scheme is applied to THE construction of Lax-type integrable discretizations of one sufficiently wide class of nonlinear integrable dynamical systems on functional manifolds. Their Lie-algebraic structure and complete integrability related to the coadjoint orbits on the Markov coalgebras is discussed. It is shown that the set of conservation laws and the associated Poisson structure can be obtained as a byproduct of the proposed approach. Based on the quasirepresentation property of Lie algebras, the limiting procedure of finding nonlinear dynamical systems on the corresponding functional spaces is demonstrated.

Article (English)

Integrability Analysis of a Two-Component Burgers-Type Hierarchy

Blackmore D. L., Özçağ E., Prikarpatskii A. K., Soltanov K. N.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2015. - 67, № 2. - pp. 147-162

The Lax integrability of a two-component polynomial Burgers-type dynamical system is analyzed by using a differential-algebraic approach. Its linear adjoint matrix Lax representation is constructed. A related recursive operator and an infinite hierarchy of nonlinear Lax integrable dynamical systems of the Burgers–Korteweg–de-Vries type are obtained by the gradient-holonomic technique. The corresponding Lax representations are presented.

Article (English)

Invariant measures for discrete dynamical systems and ergodic properties of generalized Boole-type transformations

Blackmore D. L., Golenia J., Prikarpatskii A. K., Prikarpatskii Ya. A.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2013. - 65, № 1. - pp. 44-57

Invariant ergodic measures for generalized Boole-type transformations are studied using an invariant quasimeasure generating function approach based on special solutions for the Frobenius - Perron operator. New two-dimensional Boole-type transformations are introduced, and their invariant measures and ergodicity properties are analyzed.

Article (English)

An infinite-dimensional Borsuk-Ulam-type generalization of the Leray-Schauder fixed-point theorem and some applications

Prikarpatskii A. K.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2008. - 60, № 1. - pp. 100–106

A generalization of the classical Leray-Schauder fixed-point theorem based on the infinite-dimensional Borsuk-Ulam-type antipode construction is proposed. A new nonstandard proof of the classical Leray-Schauder fixed-point theorem and a study of the solution manifold of a nonlinear Hamilton-Jacobi-type equation are presented.

Article (English)

Generalized de Rham-Hodge complexes, the related characteristic Chern classes, and some applications to integrable multidimensional differential systems on Riemannian manifolds

Bogolyubov N. N., Prikarpatskii A. K.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2007. - 59, № 3. - pp. 327–344

We study the differential-geometric aspects of generalized de Rham-Hodge complexes naturally related to integrable multidimensional differential systems of the M. Gromov type, as well as the geometric structure of the Chern characteristic classes. Special differential invariants of the Chern type are constructed, their importance for the integrability of multidimensional nonlinear differential systems on Riemannian manifolds is discussed. An example of the three-dimensional Davey-Stewartson-type nonlinear integrable differential system is considered, its Cartan type connection mapping, and related Chern-type differential invariants are analyzed.

Article (English)

Lie-algebraic structure of (2 + 1)-dimensional Lax-type integrable nonlinear dynamical systems

Hentosh О. Ye., Prikarpatskii A. K.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2004. - 56, № 7. - pp. 939–946

A Hamiltonian representation for a hierarchy of Lax-type equations on a dual space to the Lie algebra of integro-differential operators with matrix coefficients extended by evolutions for eigenfunctions and adjoint eigenfunctions of the corresponding spectral problems is obtained via some special Båcklund transformation. The connection of this hierarchy with Lax-integrable two-metrizable systems is studied.

Brief Communications (Ukrainian)

Structure of Binary Darboux-Type Transformations for Hermitian Adjoint Differential Operators

Prikarpatskii A. K., Samoilenko V. G.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2004. - 56, № 2. - pp. 271-275

For Hermitian adjoint differential operators, we consider the structure of Darboux–Bäcklund-type transformations in the class of parametrically dependent Hilbert spaces. By using the proposed new method, we obtain the corresponding integro-differential symbols of the operators of transformations in explicit form and consider the problem of their application to the construction of two-dimensional Lax-integrable nonlinear evolution equations and their Darboux–Bäcklund-type transformations.

Article (English)

Hopf Algebras and Integrable Flows Related to the Heisenberg–Weil Coalgebra

Blackmore D. L., Prikarpatskii A. K., Samoilenko A. M.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2004. - 56, № 1. - pp. 88-96

On the basis of the structure of Casimir elements associated with general Hopf algebras, we construct Liouville–Arnold integrable flows related to naturally induced Poisson structures on an arbitrary coalgebra and their deformations. Some interesting special cases, including coalgebra structures related to the oscillatory Heisenberg–Weil algebra and integrable Hamiltonian systems adjoint to them, are considered.

Article (English)

The Reduction Method in the Theory of Lie-Algebraically Integrable Oscillatory Hamiltonian Systems

Prikarpatskii A. K., Samoilenko V. G., Taneri U.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2003. - 55, № 2. - pp. 232-240

We study the problem of the complete integrability of nonlinear oscillatory dynamical systems connected, in particular, both with the Cartan decomposition of a Lie algebra \(G = K \oplus P{\text{, where }}K\) is the Lie algebra of a fixed subgroup \(K \subset {\text{G}}\) with respect to an involution σ : GG on the Lie group G, and with a Poisson action of special type on a symplectic matrix manifold.

Article (English)

Lyapunov–Schmidt Approach to Studying Homoclinic Splitting in Weakly Perturbed Lagrangian and Hamiltonian Systems

Prikarpatskii A. K., Samoilenko A. M., Samoilenko V. G.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2003. - 55, № 1. - pp. 66-74

We analyze the geometric structure of the Lyapunov–Schmidt approach to studying critical manifolds of weakly perturbed Lagrangian and Hamiltonian systems.

Article (English)

Finite-Dimensional Reductions of Conservative Dynamical Systems and Numerical Analysis. I

Brzychczy S., Prikarpatskii A. K., Samoilenko V. G.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2001. - 53, № 2. - pp. 220-228

We study infinite-dimensional Liouville–Lax integrable nonlinear dynamical systems. For these systems, we consider the problem of finding an appropriate set of initial conditions leading to typical solutions such as solitons and traveling waves. We develop an approach to the solution of this problem based on the exact reduction of a given nonlinear dynamical system to its finite-dimensional invariant submanifolds and the subsequent investigation of the system of ordinary differential equations obtained by qualitative analysis. The efficiency of the approach proposed is demonstrated by the examples of the Korteweg–de Vries equation, the modified nonlinear Schrödinger equation, and a hydrodynamic model.

Article (Ukrainian)

The Poincare-Mel'nikov geometric analysis of the transversal splitting of manifolds of slowly perturbed nonlinear dynamical systems. I

Prikarpatskii A. K., Samoilenko A. M., Timchishin O. Ya.

Full text (.pdf)

Ukr. Mat. Zh. - 1993. - 45, № 12. - pp. 1668–1681

Article (Ukrainian)

Some aspects of a gradient holonomic algorithm in the theory of integrability of nonlinear dynamic systems and computer algebra problems

Fil' B. N., Mitropolskiy Yu. A., Prikarpatskii A. K.

Full text (.pdf)

Ukr. Mat. Zh. - 1991. - 43, № 1. - pp. 78-81

Article (Ukrainian)

A bilocal periodic problem for the Sturm-Liouville and Dirac operators and some applications to the theory of nonlinear dynamical systems. I

Bogolyubov N. N., Prikarpatskii A. K.

Full text (.pdf)

Ukr. Mat. Zh. - 1990. - 42, № 6. - pp. 794–800

Article (Ukrainian)

Abelian integrals, integrable dynamic systems of the Neumann-Rosochatius type, and the Lax representation

Mikityuk I. V., Prikarpatskii A. K., Samoilenko V. G.

Full text (.pdf)

Ukr. Mat. Zh. - 1989. - 41, № 8. - pp. 1094–1100

Article (Ukrainian)

Quantum lie algebra of currents ? The universal algebraic structure of symmetries of completely integrable dynamical systems

Fil' B. N., Prikarpatskii A. K., Pritula N. N.

Full text (.pdf)

Ukr. Mat. Zh. - 1988. - 40, № 6. - pp. 764–768

Article (Ukrainian)

Algebraic scheme of discrete approximations of linear and nonlinear dynamical systems of mathematical physics

Mitropolskiy Yu. A., Prikarpatskii A. K., Samoilenko V. G.

Full text (.pdf)

Ukr. Mat. Zh. - 1988. - 40, № 4. - pp. 453-458

Article (Ukrainian)

The infinite-dimensional Schrodinger operator and its potential perturbations

Prikarpatskii A. K.

Full text (.pdf)

Ukr. Mat. Zh. - 1987. - 39, № 1. - pp. 87-92

Article (Ukrainian)

Bogolyubov's functional equation and the lie-poisson-lasov simplectic structure associated with it

Bogolyubov N. N., Prikarpatskii A. K., Samoilenko V. G.

Full text (.pdf)

Ukr. Mat. Zh. - 1986. - 38, № 6. - pp. 774–778

Article (Ukrainian)

N. N. Bogolyubov's quantum method of generating functionals in statistical physics: The current Lie algebra, its representations, and functional equations

Bogolyubov N. N., Prikarpatskii A. K.

Full text (.pdf)

Ukr. Mat. Zh. - 1986. - 38, № 3. - pp. 284–289

Article (Ukrainian)

Complete integrability of the differential equations connected with the problem of nonlinear oscillations of a homogeneous beam compressed longitudinally

Mitropolskiy Yu. A., Prikarpatskii A. K., Samoilenko V. G.

Full text (.pdf)

Ukr. Mat. Zh. - 1985. - 37, № 6. - pp. 727–729

Article (Ukrainian)

Backlund transformations for the nonlinear Korteweg-De Vries equation from the algebrogeometric point of view

Prikarpatskii A. K., Samoilenko V. G.

Full text (.pdf)

Ukr. Mat. Zh. - 1985. - 37, № 5. - pp. 670–673

Article (Ukrainian)

Integrability of ideals in Grassman algebras on differentiable manifolds and some of its applications

Mitropolskiy Yu. A., Prikarpatskii A. K., Samoilenko V. G.

Full text (.pdf)

Ukr. Mat. Zh. - 1984. - 36, № 4. - pp. 451 – 456

Article (Ukrainian)

Periodic problem for a Toda chain

Prikarpatskii A. K., Samoilenko V. G.

Full text (.pdf)

Ukr. Mat. Zh. - 1982. - 34, № 4. - pp. 469—475

Article (Ukrainian)

Geometric structure and Backlund transformation of a system of nonlinear partial differential evolution equations

Prikarpatskii A. K.

Full text (.pdf)

Ukr. Mat. Zh. - 1980. - 32, № 1. - pp. 124 - 128

Article (Ukrainian)

A system of nonlinear differential equations with an exact solution

Prikarpatskii A. K.

Full text (.pdf)

Ukr. Mat. Zh. - 1979. - 31, № 5. - pp. 576–582

Article (Ukrainian)

Periodic problem for the classical two-dimensional Thirring model

Golod P. I., Prikarpatskii A. K.

Full text (.pdf)

Ukr. Mat. Zh. - 1979. - 31, № 4. - pp. 454–459