Maria Demina
31 Kashirskoe Shosse, 115409 Moscow, Russian Federation
Department of Applied Mathematics,
National Research Nuclear University “MEPhI”
Publications:
Safonova D. V., Demina M. V., Kudryashov N. A.
Stationary Configurations of Point Vortices on a Cylinder
2018, vol. 23, no. 5, pp. 569579
Abstract
In this paper we study the problem of constructing and classifying stationary
equilibria of point vortices on a cylindrical surface. Introducing polynomials with roots at vortex
positions, we derive an ordinary differential equation satisfied by the polynomials. We prove
that this equation can be used to find any stationary configuration. The multivortex systems
containing point vortices with circulation $\Gamma_1$ and $\Gamma_2$ $(\Gamma_2 = \mu\Gamma_1)$ are considered in detail.
All stationary configurations with the number of point vortices less than five are constructed.
Several theorems on existence of polynomial solutions of the ordinary differential equation under
consideration are proved. The values of the parameters of the mathematical model for which
there exists an infinite number of nonequivalent vortex configurations on a cylindrical surface
are found. New point vortex configurations are obtained.

Demina M. V., Kudryashov N. A.
Multiparticle Dynamical Systems and Polynomials
2016, vol. 21, no. 3, pp. 351366
Abstract
Polynomial dynamical systems describing interacting particles in the plane are studied. A method replacing integration of a polynomial multiparticle dynamical system
by finding polynomial solutions of partial differential equations is introduced. The method enables one to integrate a wide class of polynomial multiparticle dynamical systems. The general solutions of certain dynamical systems related to linear secondorder partial differential equations are found. As a byproduct of our results, new families of orthogonal polynomials are derived.

Demina M. V., Kudryashov N. A.
Relative Equilibrium Configurations of Point Vortices on a Sphere
2013, vol. 18, no. 4, pp. 344355
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.

Demina M. V., Kudryashov N. A.
Point Vortices and Classical Orthogonal Polynomials
2012, vol. 17, no. 5, pp. 371384
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.

Demina M. V., Kudryashov N. A.
Point Vortices and Polynomials of the Sawada–Kotera and Kaup–Kupershmidt Equations
2011, vol. 16, no. 6, pp. 562576
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.
