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2013
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# Maria Demina

31 Kashirskoe Shosse, 115409 Moscow, Russian Federation
Department of Applied Mathematics, National Research Nuclear University “MEPhI”

## Publications:

 Demina M. V., Kudryashov N. A. Multi-particle Dynamical Systems and Polynomials 2016, vol. 21, no. 3, pp.  351-366 Abstract Polynomial dynamical systems describing interacting particles in the plane are studied. A method replacing integration of a polynomial multi-particle dynamical system by finding polynomial solutions of partial differential equations is introduced. The method enables one to integrate a wide class of polynomial multi-particle dynamical systems. The general solutions of certain dynamical systems related to linear second-order partial differential equations are found. As a by-product of our results, new families of orthogonal polynomials are derived. Keywords: multi-particle dynamical systems, polynomial solutions of partial differential equations, orthogonal polynomials Citation: Demina M. V., Kudryashov N. A.,  Multi-particle Dynamical Systems and Polynomials, Regular and Chaotic Dynamics, 2016, vol. 21, no. 3, pp. 351-366 DOI:10.1134/S1560354716030072
 Demina M. V., Kudryashov N. A. Relative Equilibrium Configurations of Point Vortices on a Sphere 2013, vol. 18, no. 4, pp.  344-355 Abstract The problem of constructing and classifying equilibrium and relative equilibrium configurations of point vortices on a sphere is studied. A method which enables one to find any such configuration is presented. Configurations formed by the vortices placed at the vertices of Platonic solids are considered without making the assumption that the vortices possess equal in absolute value circulations. Several new configurations are obtained. Keywords: point vortices, sphere, relative equilibrium, fixed equilibrium, Platonic solids Citation: Demina M. V., Kudryashov N. A.,  Relative Equilibrium Configurations of Point Vortices on a Sphere, Regular and Chaotic Dynamics, 2013, vol. 18, no. 4, pp. 344-355 DOI:10.1134/S1560354713040023
 Demina M. V., Kudryashov N. A. Point Vortices and Classical Orthogonal Polynomials 2012, vol. 17, no. 5, pp.  371-384 Abstract Stationary equilibria of point vortices in the plane and on the cylinder in the presence of a background flow are studied. Vortex systems with an arbitrary choice of circulations are considered. Differential equations satisfied by generating polynomials of vortex configurations are derived. It is shown that these equations can be reduced to a single one. It is found that polynomials that are Wronskians of classical orthogonal polynomials solve the latter equation. As a consequence vortex equilibria at a certain choice of background flows can be described with the help of Wronskians of classical orthogonal polynomials. Keywords: point vortices, special polynomials, classical orthogonal polynomials Citation: Demina M. V., Kudryashov N. A.,  Point Vortices and Classical Orthogonal Polynomials, Regular and Chaotic Dynamics, 2012, vol. 17, no. 5, pp. 371-384 DOI:10.1134/S1560354712050012
 Demina M. V., Kudryashov N. A. Point Vortices and Polynomials of the Sawada–Kotera and Kaup–Kupershmidt Equations 2011, vol. 16, no. 6, pp.  562-576 Abstract Rational solutions and special polynomials associated with the generalized $K_2$ hierarchy are studied. This hierarchy is related to the Sawada–Kotera and Kaup–Kupershmidt equations and some other integrable partial differential equations including the Fordy–Gibbons equation. Differential–difference relations and differential equations satisfied by the polynomials are derived. The relationship between these special polynomials and stationary configurations of point vortices with circulations $\Gamma$ and $−2\Gamma$ is established. Properties of the polynomials are studied. Differential–difference relations enabling one to construct these polynomials explicitly are derived. Algebraic relations satisfied by the roots of the polynomials are found. Keywords: point vortices, special polynomials, generalized $K_2$ hierarchy, Sawada–Kotera equation, Kaup–Kupershmidt equation, Fordy–Gibbons equation Citation: Demina M. V., Kudryashov N. A.,  Point Vortices and Polynomials of the Sawada–Kotera and Kaup–Kupershmidt Equations, Regular and Chaotic Dynamics, 2011, vol. 16, no. 6, pp. 562-576 DOI:10.1134/S1560354711060025