The generalized Yablonskii–Vorob'ev polynomials and their properties

Kudryashov, Nikolay A.; Demina, Maria V.

doi:10.1016/j.physleta.2008.04.069

Cited by 16 publications

(8 citation statements)

References 27 publications

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“…Thus, Clarkson & Mansfield (2003) have exhibited highly symmetric 'triangular' confinement regions in the complex plane for the standard Yablonskii-Vorob'ev polynomials. This elegant work has most recently been followed by consideration of generalized Yablonskii-Vorob'ev polynomials and the Painlevé II hierarchy by Kudryashov & Demina (2007, 2008.…”

Section: Introductionmentioning

confidence: 99%

Electrical structures of interfaces: a Painlevé II model

et al. 2010

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A Painlevé II model derived out of the classical Nernst-Planck system is applied in the context of boundary value problems that describe the electric field distribution in a region x > 0 occupied by an electrolyte. For privileged flux ratios of the ion concentrations, the auto-Bäcklund transformation admitted by the Painlevé II equation may be applied iteratively to construct exact solutions to classes of physically relevant boundary value problems. These representations involve, in turn, either Yablonskii-Vorob'ev polynomials or classical Airy functions. The requirement that the electric field distribution and ion concentrations in these representations be non-singular imposes constraints on the physical parameters. These are investigated in detail along with asymptotic properties.

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Section: Introductionmentioning

confidence: 99%

Electrical structures of interfaces: a Painlevé II model

et al. 2010

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“…As a result the Wronskian representation arises. Some other recurrence relations satisfied by these polynomials are given in [10,[15][16][17][18]. Two successive Adler -Moser polynomials solve the Tkachenko equation, i. e. we may setP (z) = V k±1 (z),Q(z) = V k (z) in (13).…”

Section: Properties Of the Tkachenko Equation And The Generalized Tkamentioning

confidence: 99%

“…., N by means of equation ( 13). Suppose we have found a solution of equation ( 13) in the form (16), then a vortex with circulation Γ j is situated at the point z = z 0 whenever the function P (z) has a "root" of "multiplicity" Γ j (Γ j + 1)/2 at the point z = z 0 and the function Q(z) has a "root" of "multiplicity" Γ j (Γ j − 1)/2 at the point z = z 0 . The circulation is calculated as the difference of the corresponding "multiplicities".…”

Section: Stationary Equilibria Of Point Vorticesmentioning

confidence: 99%

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

Demina¹,

Kudryashov²

2012

J. Phys. A: Math. Theor.

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Stationary and translating relative equilibria of point vortices in the plane are studied. It is shown that stationary equilibria of a system containing point vortices with arbitrary choice of circulations can be described with the help of the Tkachenko equation. It is obtained that the Adler -Moser polynomial are not unique polynomial solutions of the Tkachenko equation. A generalization of the Tkachenko equation to the case of translating relative equilibria is derived. It is shown that the generalization of the Tkachenko equation possesses polynomial solutions with degrees that are not triangular numbers. IntroductionThe problem of finding stationary and relative equilibrium solutions to Helmholtz's equations describing motion of point vortices in the plane has been attracting much attention during recent years [1][2][3][4][5][6][7][8][9]. The so-called "polynomial method" is often used while studying this problem [1]. According to this method one introduces polynomials with roots at vortex positions. For example, it was found by Stieltjes that the generating polynomial of N identical point vortices on a line is essentially the Nth Hermite polynomial. Considering equilibrium of point vortices with equal in absolute value circulations leads to an ordinary differential equation first discovered by Tkachenko [1]. It is well known that the Adler -Moser polynomials give polynomial solutions to the Tkachenko equation. Originally these polynomials arose in 1 the theory of one of the most famous soliton equation, the Korteweg -de Vries equation [10]. This fact provides a remarkable and rather unexpected connection between the dynamics of point vortices and the theory of integrable partial differential equations [11]. Not long ago analogous connections between equilibria of point vortices with circulations Γ, −2Γ and rational solutions of the Sawada -Kotera and the the Kaup -Kupershmidt equations was established [12].In this article we study stationary and translating relative equilibria of multivortex systems with circulations Γ 1 , . . ., Γ N . Our aim is to show that the stationary case can be described with the help of the Tkachenko equation and the translating case is reducible to a generalization of the Tkachenko equation. As a consequence of our results we obtain that the Adler -Moser polynomials are not unique polynomial solutions of the Tkachenko equation.This article is organized as follows. In section 2 we consider stationary equilibria of point vortices with arbitrary choice of circulations. We derive an ordinary differential equation satisfied by generating polynomials of arrangements and transform this equation to the Tkachenko equation. Section 3 is devoted to the translating case. We find an ordinary differential equation satisfied by generating polynomials of the vortices and reduce the resulting equation to the generalization of the Tkachenko equation. In section 4 we study properties of the Tkachenko equation and the generalized Tkachenko equation. Quite unexpectedly it is possible to constr...

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“…According to this method polynomials with roots at vortex positions are introduced. This approach provides quite unexpected connection between dynamics of point vortices and the theory of classical and nonlinear special polynomials [2,12,13,[15][16][17][18]. For example, the generating polynomial of M identical point vortices on a line in rotating relative equilibrium is essentially the Mth Hermite polynomial.…”

Section: Introductionmentioning

confidence: 99%

Point vortices and classical orthogonal polynomials

Demina

Kudryashov

2012

Regul. Chaot. Dyn.

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Stationary equilibria of point vortices with arbitrary choice of circulations in a background flow are studied. 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.

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The generalized Yablonskii–Vorob'ev polynomials and their properties

Cited by 16 publications

References 27 publications

Electrical structures of interfaces: a Painlevé II model

Electrical structures of interfaces: a Painlevé II model

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

Point vortices and classical orthogonal polynomials

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