# Prykarpatsky A. K.

### Theory of multidimensional Delsarte – Lions transmutation operators. II

Blackmore D., Prykarpatsky A. K., Prykarpatsky Ya. A., Samoilenko A. M.

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 6. - pp. 808-839

UDC 517.9

The differential-geometric and topological structures related to the Delsarte transmutation operators and the Gelfand – Levitan – Marchenko equations that describe these operators are studied by using sutable differential de Rham – Hodge – Skrypnik complexes.
The correspondence between the spectral theory and special Berezansky-type congruence properties of the Delsarte transmutation operators is established. Some applications to multidimensional differential operators are presented, including the three-dimensional Laplace operator, the two-dimensional classical Dirac operator, and its multidimensional affine extension associated with self-dual Yang – Mills equations.
Soliton solutions of a certain class of dynamical systems are discussed.

### Theory of multidimensional Delsarte – Lions transmutation operators. I

Blackmore D., Prykarpatsky A. K., Prykarpatsky Ya. A., Samoilenko A. M.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 12. - pp. 1660-1695

We present a brief review of the original results obtained by the authors in the theory of Delsarte –Lions transmutations of multidimensional spectral differential ope rators based on the classical works by Yu. M. Berezansky, V. A. Marchenko, B. M. Levitan, and R. G. Newton, on the well-known L. D. Faddeev’s survey, the book by L. P. Nyzhnyk, and the generalized De-Rham – Hodge theory suggested by I. V. Skrypnik and developed by the authors for the differential-operator complexes. The operator structure of Delsarte – Lions transformations and the properties of their Volterra factorizations are analyzed in detail. In particular, we study the differential-geometric and topological structures of the spectral properties of the Delsarte – Lions transmutations within the framework of the generalized De-Rham – Hodge theory.

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

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.

### Integrability Analysis of a Two-Component Burgers-Type Hierarchy

Blackmore D., Özçağ E., Prykarpatsky A. K., Soltanov K. N.

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.

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

Blackmore D., Golenia J., Prykarpatsky A. K., Prykarpatsky Ya. A.

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.

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

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.

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

Bogolyubov N. N., Prykarpatsky A. K.

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.

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

Hentosh О. Ye., Prykarpatsky A. K.

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.

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

Prykarpatsky A. K., Samoilenko V. G.

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.

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

Blackmore D., Prykarpatsky A. K., Samoilenko A. M.

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.

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

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

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 : *G* → *G* on the Lie group *G*, and with a Poisson action of special type on a symplectic matrix manifold.

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

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

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.

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

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

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.

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

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

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

On the basis of the geometric ideas of Poincaré and Mel'nikov, we study sufficient criteria of the transversal splitting of heteroclinic separatrix manifolds of slowly perturbed nonlinear dynamical systems with a small parameter. An example of adiabatic invariance breakdown is considered for a system on a plane.

### An issue dedicated to the illustrious memory of Mykol Mykolayovych Bogolyubov

Bogolyubov N. N., Mitropolskiy Yu. A., Prykarpatsky A. K., Rudavsky Yu.K., Samoilenko A. M., Vakarchuk I. V.

Ukr. Mat. Zh. - 1992. - 44, № 9. - pp. 1155-1156

### Reduction and geometric quantization

Mikityuk I. V., Prykarpatsky A. K.

Ukr. Mat. Zh. - 1992. - 44, № 9. - pp. 1220–1228

A construction is created that makes it possible to geometrically quantize a reduced Hamiltonian system using the procedure of geometric quantization realized for a Hamiltonian system with symmetries (i.e., to find the discrete spectrum and the corresponding eigenfunctions, if these have been found for the initial system). The construction is used to geometrically quantize a system obtained by reduction of a Hamiltonian system that determines the geodesic flow on an $n$-dimensional sphere.

### Category of topological jet manifolds and certain applications in the theory of nonlinear infinite-dimensional dynamical systems

Fil' B. N., Prykarpatsky A. K.

Ukr. Mat. Zh. - 1992. - 44, № 9. - pp. 1242–1256

A new category of topological jet manifolds is proposed for the purpose of investigating exact finite-dimensional approximations of nonlinear dynamical systems on infinite-dimensional functional manifolds. Differential geometry structures on these manifolds and their applications to the theory of integrability in quadratures of nonlinear dynamical Lax-type systems are studied.

### 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., Prykarpatsky A. K.

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

### 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., Prykarpatsky A. K.

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

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

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

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

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

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

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

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

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

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

### The infinite-dimensional Schrodinger operator and its potential perturbations

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

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

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

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

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

Bogolyubov N. N., Prykarpatsky A. K.

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

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

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

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

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

Prykarpatsky A. K., Samoilenko V. G.

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

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

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

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

### Periodic problem for a Toda chain

Prykarpatsky A. K., Samoilenko V. G.

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

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

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

### A system of nonlinear differential equations with an exact solution

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

### Periodic problem for the classical two-dimensional Thirring model

Golod P. I., Prykarpatsky A. K.

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