arXiv subjects

Alexey A. Magazev

Publications and source records attributed to Alexey A. Magazev.

6 recordsLinked to original sources

arXiv2020

Classification of scalar second-order differential equations with low-dimensional symmetry groups: The case of free action

Based on an original classification of differential equations by types of regular Lie group actions, we offer a systematic procedure for describing partial differential equations with prescribed symmetry groups. Using a new powerful algebraic technique based on the so-called covariant form of a differential equation, we give an effective algorithm for constructing differential equations whose symmetry groups regularly and freely act on the space of dependent and independent variables. As an application, we derive a complete classification of quasi-linear scalar second-order partial differential equations with regular free symmetry groups of dimension no greater than three.

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arXiv2020

Exact solutions of the Klein-Gordon equation in external electromagnetic fields on 3D de Sitter background

In this study, we investigate the symmetry properties and the possibility of exact integration of the Klein--Gordon equation in the presence of an external electromagnetic field on 3D de Sitter background. We present an algorithm for constructing the first-order symmetry algebra and describe its structure in terms of Lie algebra extensions. Based on the well-known classification of the inequivalent subalgebras of the algebra $\mathfrak{so}(1,3)$, we obtain the classification of the electromagnetic fields on $\mathrm{dS}_3$ admitting first-order symmetry algebras of the Klein-Gordon equation. Then, we select the integrable cases, and for each of them, we construct exact solutions, using the non-commutative integration method developed by Shapovalov and Shirokov. In Appendix, we present an original algebraic method for constructing the special local coordinates on de Sitter space, in which the basis vector fields for subalgebras of the algebra $\mathfrak{so}(1,3)$ have the simplest form.

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arXiv2015

Computation of Composition Functions and Invariant Vector Fields in Terms of Structure Constants of Associated Lie Algebras

Methods of construction of the composition function, left- and right-invariant vector fields and differential 1-forms of a Lie group from the structure constants of the associated Lie algebra are proposed. It is shown that in the second canonical coordinates these problems are reduced to the matrix inversions and matrix exponentiations, and the composition function can be represented in quadratures. Moreover, it is proven that the transition function from the first canonical coordinates to the second canonical coordinates can be found by quadratures.

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arXiv2015

A method of integration for classical and quantum equations based on the connection between canonical transformations and irreducible representations of Lie groups

We propose a method for integrating the right-invariant geodesic flows on Lie groups based on the use of a special canonical transformation in the cotangent bundle of the group. We also describe an original method of constructing exact solutions for the Klein - Gordon equation on unimodular Lie groups. Finally, we formulate a theorem which establishes a connection between the special canonical transformation and irreducible representations of Lie group. This connection allows us to consider the proposed methods of integrating for classical and quantum equations in the framework of a unified approach.

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arXiv2014

Integrating Klein-Gordon-Fock equations in an external electromagnetic field on Lie groups

We investigate the structure of the Klein-Gordon-Fock equation symmetry algebra on pseudo-Riemannian manifolds with motions in the presence of an external electromagnetic field. We show that in the case of an invariant electromagnetic field tensor, this algebra is a one-dimensional central extension of the Lie algebra of the group of motions. Based on the coadjoint orbit method and harmonic analysis on Lie groups, we propose a method for integrating the Klein-Gordon-Fock equation in an external field on manifolds with simply transitive group actions. We consider a nontrivial example on the four-dimensional group $E(2) \times \mathbb{R}$ in detail.

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arXiv2011

Integrable magnetic geodesic flows on Lie groups

Right-invariant geodesic flows on manifolds of Lie groups associated with 2-cocycles of corresponding Lie algebras are discussed. Algebra of integrals of motion for magnetic geodesic flows is considered and necessary and sufficient condition of integrability in quadratures is formulated. Canonic forms for 2-cocycles of all 4-dimensional Lie algebras are given and integrable cases among them are separated.

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