arXiv subjects

Anatolij K. Prykarpatski

Publications and source records attributed to Anatolij K. Prykarpatski.

7 recordsLinked to original sources

arXiv2023

The Courant type algebroids, the coadjoint orbits and related integrable flows

Poisson structures related with the affine Courant type algebroid are analyzed, including \ those related with cotangent bundles on Lie group manifolds. A special attantion is paid to Courant type algebroids and related R-structures \ on them, generated by suitably defined tensor mappings. \ There are constructed Lie-Poisson brackets invariant with respect to the coadjoint action of the loop diffeomorphisms group and described the related Courant type algebroids. \ The corresponding integrable Hamiltonian flows, generated by Casimir functionals and generalizing the so called heavenly type differential systems, describing diverse geometric structures of conformal type on finite dimensional Rieamnnian manifolds are described.

math.SG↗

arXiv2019

Dispersionless Multi-Dimensional Integrable Systems and Related Conformal Structure Generating Equations of Mathematical Physics

Using diffeomorphism group vector fields on $\mathbb{C}$-multiplied tori and the related Lie-algebraic structures, we study multi-dimensional dispersionless integrable systems that describe conformal structure generating equations of mathematical physics. An interesting modification of the devised Lie-algebraic approach subject to spatial-dimensional invariance and meromorphicity of the related differential-geometric structures is described and applied in proving complete integrability of some conformal structure generating equations. As examples, we analyze the Einstein-Weyl metric equation, the modified Einstein-Weyl metric equation, the Dunajski heavenly equation system, the first and second conformal structure generating equations and the inverse first Shabat reduction heavenly equation. We also analyze the modified Plebański heavenly equations, the Husain heavenly equation and the general Monge equation along with their multi-dimensional generalizations. In addition, we construct superconformal analogs of the Whitham heavenly equation.

math-ph↗

arXiv2018

On the Linearization Covering Technique and its Application to Integrable Nonlinear Differential Systems

In this letter I analyze a covering jet manifold scheme, its relation to the invariant theory of the associated vector fields, and applications to the Lax-Sato-type integrability of nonlinear dispersionless differential systems. The related contact geometry linearization covering scheme is also discussed. The devised techniques are demonstrated for such nonlinear Lax-Sato integrable equations as Gibbons-Tsarev, ABC, Manakov-Santini and the differential Toda singular manifold equations.

nlin.SI↗

arXiv2015

On the completely integrable Calogero type discretizations of nonlinear Lax integrable dynamical systems and the related Markov type co-adjoint orbits

The Calogero type matrix discretization scheme is applied to constructing the Lax type integrable discretizations of one wide enough class of nonlinear integrable dynamical systems on functional manifolds. Their Lie-algebraic structure and complete integrability related with co-adjoint orbits on the Markov co-algebras is discussed. It is shown that a set of conservation laws and the associated Poisson structure ensue as a byproduct of the approach devised. Based on the Lie algebras quasi-representation property the limiting procedure of finding the nonlinear dynamical systems on the corresponding functional spaces is demonstrated.

math-ph↗

arXiv2014

Discrete approximations on functional classes for the integrable nonlinear Schrödinger dynamical system: a symplectic finite-dimensional reduction approach

We investigate discretizations of the integrable discrete nonlinear Schrödinger dynamical system and related symplectic structures. We develop an effective scheme of invariant reducing the corresponding infinite system of ordinary differential equations to an equivalent finite system of ordinary differential equations with respect to the evolution parameter. We construct a finite set of recurrent algebraic regular relations allowing to generate solutions of the discrete nonlinear Schrödinger dynamical system and we discuss the related functional spaces of solutions. Finally, we discuss the Fourier transform approach to studying the solution set of the discrete nonlinear Schrödinger dynamical system and its functional-analytical aspects.

nlin.SI↗

arXiv2013

The Lax integrability of a two-component hierarchy of the Burgers type dynamical systems within asymptotic and differential-algebraic approaches

The Lax type integrability of a two-component polynomial Burgers type dynamical system within a differential-algebraic approach is studied, its linear adjoint matrix Lax representation is constructed. A related recursion operator and infnite hierarchy of Lax integrable nonlinear dynamical systems of the Burgers-Korteweg-de Vries type are derived by means of the gradient-holonomic technique, the corresponding Lax type representations are presented.

nlin.SI↗

arXiv2013

Differential-algebraic approach to constructing representations of commuting differentiations in functional spaces and its application to nonlinear integrable dynamical systems

There is developed a differential-algebraic approach to studying the representations of commuting differentiations in functional differential rings under nonlinear differential constraints. An example of the differential ideal with the only one conserved quantity is analyzed in detail, the corresponding Lax type representations of differentiations are constructed for an infinite hierarchy of nonlinear dynamical systems of the Burgers and Korteweg-de Vries type. A related infinite bi-Hamiltonian hierarchy of Lax type dynamical systems is constructed.

nlin.SI↗