Maria V. Demina

Traveling wave solutions of the generalized nonlinear evolution equations

Solitary wave solutions for a family of nonlinear evolution equations with an arbitrary parameter in the exponents are constructed. Some of the obtained solutions seem to be new.

On elliptic solutions of nonlinear ordinary differential equations

The problem of constructing and classifying exact elliptic solutions of autonomous nonlinear ordinary differential equations is studied. An algorithm for finding elliptic solutions in explicit form is presented.

Point vortices and polynomials of the Sawada-Kotera and Kaup-Kupershmidt equations

Rational solutions and special polynomials associated with the generalized K2 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 Γ and −2Γ 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.

The Yablonskii–Vorob’ev polynomials for the second Painlevé hierarchy

Abstract Special polynomials associated with rational solutions of the second Painleve equation and other equations of its hierarchy are studied. A new method, which allows one to construct each family of polynomials is presented. The structure of the polynomials is established. Formulae for their coefficients are found. The degree of every polynomial is obtained. The main achievement of the method lies in the fact that it enables one to construct the family of polynomials corresponding to any member of the second Painleve hierarchy. Our approach can be applied for deriving the polynomials related to rational or algebraic solutions of other nonlinear differential equations.

Vortices and polynomials: non-uniqueness of the Adler–Moser polynomials for the Tkachenko equation

Stationary and translating relative equilibria of point vortices in the plane are studied. It is shown that stationary equilibria of any system containing point vortices with arbitrary choice of circulations can be described with the help of the Tkachenko equation. It is also obtained that translating relative equilibria of point vortices with arbitrary circulations can be constructed using a generalization of the Tkachenko equation. Roots of any pair of polynomials solving the Tkachenko equation and the generalized Tkachenko equation are proved to give positions of point vortices in stationary and translating relative equilibria accordingly. These results are valid even if the polynomials in a pair have multiple or common roots. It is obtained that the Adler–Moser polynomial provides non-unique polynomial solutions of the Tkachenko equation. It is shown that the generalized Tkachenko equation possesses polynomial solutions with degrees that are not triangular numbers.

Special polynomials associated with the fourth order analogue to the Painlevé equations

Abstract Rational solutions of the fourth order analogue to the Painleve equations are classified. Special polynomials associated with the rational solutions are introduced. The structure of the polynomials is found. Formulae for their coefficients and degrees are derived. It is shown that special solutions of the Fordy–Gibbons, the Caudrey–Dodd–Gibbon and the Kaup–Kupershmidt equations can be expressed through solutions of the equation studied.

The polygonal method for constructing exact solutions to certain nonlinear differential equations describing water waves

A method is proposed for constructing exact solutions to certain nonlinear differential equations of mathematical physics. Possible applications of this method are illustrated using equations arising in the description of water waves. Exact solutions to the generalized Gardner, Kawahara, and Benjamin-Bona-Mahony equations are constructed.

Point vortices and classical orthogonal polynomials

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.

Relative equilibrium configurations of point vortices on a sphere

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.

Polynomial Method for Constructing Equilibrium Configurations of Point Vortices in a Plane

The problem of constructing and classifying stationary and translating configurations of point vortices with an arbitrary choice of circulations is studied. The polynomial method enabling one to find any such configuration is described in detail. Stationary configurations for vortex systems with circulations Γ, −μΓ are classified in the case of integer μ. New configurations are obtained.