Prykarpatsky Ya. A.
Ukr. Mat. Zh. - 2019. - 71, № 6. - pp. 808-839
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.
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.
Differential-geometric structure and the Lax – Sato integrability of a class of dispersionless heavenly type equations
Ukr. Mat. Zh. - 2018. - 70, № 2. - pp. 293-297
This short communication is devoted to the study of differential-geometric structure and the Lax – Sato integrability of the reduced Shabat-type, Hirota, and Kupershmidt heavenly equations.
The classical M. A. Buhl problem, its Pfeiffer – Sato solutions and the classical Lagrange – D’Alembert principle for the integrable heavenly type nonlinear equations
Ukr. Mat. Zh. - 2017. - 69, № 12. - pp. 1652-1689
The survey is devoted to old and recent investigations of the classical M. A. Buhl problem of description of the compatible linear vector field equations and their general M. G. Pfeiffer and modern Lax – Sato-type special solutions. In particular, we analyze the related Lie-algebraic structures and the properties of integrability for a very interesting class of nonlinear dynamical systems called the dispersion-free heavenly type equations, which were introduced by Pleba´nski and later analyzed in a series of articles. The AKS-algebraic and related \scrR -structure schemes are used to study the orbits of the corresponding coadjoint actions, which are intimately connected with the classical Lie – Poisson structures on them. It is shown that their compatibility condition coincides with the corresponding heavenly type equations under consideration. It is also demonstrated that all these equations are originated in this way and can be represented as a Lax compatibility condition for specially constructed loop vector fields on the torus. The infinite hierarchy of conservations laws related to the heavenly equations is described and its analytic structure connected with the Casimir invariants is indicated. In addition, we present typical examples of equations of this kind demonstrating in detail their integrability via the scheme proposed in the paper. The relationship between a very interesting Lagrange – d’Alembert-type mechanical interpretation of the devised integrability scheme and the Lax – Sato equations is also discussed.
Invariant measures for discrete dynamical systems and ergodic properties of generalized Boole-type transformations
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.
Hamiltonian geometric connection associated with adiabatically perturbed Hamiltonian systems and the existence of adiabatic invariants
Ukr. Mat. Zh. - 2008. - 60, № 3. - pp. 382–387
Differential-geometric properties of the Hamiltonian connections on symplectic manifolds for the adiabatically perturbed Hamiltonian system are studied. Namely, the associated Hamiltonian connection on the main foliation is constructed and its description is given in terms of covariant derivatives and the curvature form of the corresponding connection.
Ukr. Mat. Zh. - 2006. - 58, № 6. - pp. 787–803
We develop a symplectic method for the investigation of invariant submanifolds of nonautonomous Hamiltonian systems and ergodic measures on them. The so-called Mel’nikov-Samoilenko problem for the case of adiabatically perturbed completely integrable oscillator-type Hamiltonian systems is studied on the basis of a new construction of “ virtual” canonical transformations.
Symplectic method for the construction of ergodic measures on invariant submanifolds of nonautonomous hamiltonian systems: Lagrangian manifolds, their structure, and mather homologies
Ukr. Mat. Zh. - 2006. - 58, № 5. - pp. 675–691
We develop a new approach to the study of properties of ergodic measures for nonautonomous periodic Hamiltonian flows on symplectic manifolds, which are used in many problems of mechanics and mathematical physics. Using Mather’s results on homologies of invariant probability measures that minimize some Lagrangian functionals and the symplectic theory developed by Floer and others for the investigation of symplectic actions and transversal intersections of Lagrangian manifolds, we propose an analog of a Mather-type ?-function for the study of ergodic measures associated with nonautonomous Hamiltonian systems on weakly exact symplectic manifolds. Within the framework of the Gromov-Salamon-Zehnder elliptic methods in symplectic geometry, we establish some results on stable and unstable manifolds for hyperbolic invariant sets, which are used in the theory of adiabatic invariants of slowly perturbed integrable Hamiltonian systems.
Ukr. Mat. Zh. - 2003. - 55, № 12. - pp. 1704-1719
On the basis of generalized Lagrange identity for pairs of formally adjoint multidimensional differential operators and a special differential geometric structure associated with this identity, we propose a general scheme of the construction of corresponding transformation operators that are described by nontrivial topological characteristics. We construct explicitly the corresponding integro-differential symbols of transformation operators, which are used in the construction of Lax-integrable nonlinear two-dimensional evolutionary equations and their Darboux–Bäcklund-type transformations.
Investigation of invariant deformations of integral manifolds of adiabatically perturbed completely integrable Hamiltonian systems. II
Ukr. Mat. Zh. - 1999. - 51, № 11. - pp. 1513–1528
By using the Cartan differential-geometric theory of integral submanifolds (invariant tori) of completely Liouville—Arnold integrable Hamiltonian systems on the cotangent phase space, we consider an algebraic-analytical method for the investigation of the corresponding mapping of imbedding of an invariant torus into the phase space. This enables one to describe analytically the structure of quasiperiodic solutions of the Hamiltonian system under consideration. We also consider the problem of existence of adiabatic invariants associated with a slowly perturbed Hamiltonian system.
Investigation of invariant deformations of integral manifolds of adiabatically perturbed completely integrable hamiltonian systems. I
Ukr. Mat. Zh. - 1999. - 51, № 10. - pp. 1379–1390
By using the Cartan differential-geometric theory of integral submanifolds (invariant tori) of completely Liouville-Arnol’d integrable Hamiltonian systems on the cotangent phase space, we consider an algebraic-analytic method for the investigation of the corresponding mapping of imbedding of an invariant torus into the phase space. This enables one to describe analytically the structure of quasiperiodic solutions of the Hamiltonian system under consideration.