Point vortices and classical orthogonal polynomials

Demina, Maria V.; Kudryashov, Nikolay A.

doi:10.1134/s1560354712050012

Cited by 12 publications

(12 citation statements)

References 23 publications

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“…Moreover, comparing (2.17) with (2.5) and (2.16) we conclude that the zeros of consecutive Adler-Moser polynomials are stationary configurations (or equivalently, μ = (μ 1 , μ 2 ) defined in (2.13) is a critical vector measure) for the vector energy E ϕ,a ( μ) defined in (2.14), with ϕ ≡ (0, 0) and a = − 1 2 , fact that was already observed in [17]. Further generalizations of these ideas, and in particular, their relation to rational solutions of Painlevé equations, can be found in [15,16,30,34,35,37,38,57,62,78], to cite a few.…”

Section: Definition 22mentioning

confidence: 74%

Electrostatic Partners and Zeros of Orthogonal and Multiple Orthogonal Polynomials

Martínez–Finkelshtein

Orive

Sánchez-Lara

2022

Constr Approx

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For a given polynomial P with simple zeros, and a given semiclassical weight w, we present a construction that yields a linear second-order differential equation (ODE), and in consequence, an electrostatic model for zeros of P. The coefficients of this ODE are written in terms of a dual polynomial that we call the electrostatic partner of P. This construction is absolutely general and can be carried out for any polynomial with simple zeros and any semiclassical weight on the complex plane. An additional assumption of quasi-orthogonality of P with respect to w allows us to give more precise bounds on the degree of the electrostatic partner. In the case of orthogonal and quasi-orthogonal polynomials, we recover some of the known results and generalize others. Additionally, for the Hermite–Padé or multiple orthogonal polynomials of type II, this approach yields a system of linear second-order differential equations, from which we derive an electrostatic interpretation of their zeros in terms of a vector equilibrium. More detailed results are obtained in the special cases of Angelesco, Nikishin, and generalized Nikishin systems. We also discuss the discrete-to-continuous transition of these models in the asymptotic regime, as the number of zeros tends to infinity, into the known vector equilibrium problems. Finally, we discuss how the system of obtained second-order ODEs yields a third-order differential equation for these polynomials, well described in the literature. We finish the paper by presenting several illustrative examples.

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Section: Definition 22mentioning

confidence: 74%

Electrostatic Partners and Zeros of Orthogonal and Multiple Orthogonal Polynomials

Martínez–Finkelshtein

Orive

Sánchez-Lara

2022

Constr Approx

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“…., a M satisfies equation Further let us study some other interesting examples involving classical orthogonal polynomials. It was proved in article [21] that the Wronskians P (z) = W [p i 1 , . .…”

Section: )mentioning

confidence: 99%

“…In the latter case one should divide the particles into groups according to the values of mass, charge, circulation etc. and introduce polynomials for each group separately [19][20][21][22][23][24]. Along with this it can be seen that polynomials with multiple roots satisfying partial differential equations give rise to dynamical systems describing behavior of distinct particles.…”

Section: Polynomial Multi-particle Dynamical Systemsmentioning

confidence: 99%

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Multi-particle dynamical systems and polynomials

Demina

Kudryashov

2016

Regul. Chaot. Dyn.

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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 a partial differential equations is described. 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. Our approach is also applicable to dynamical systems that are not multi-particle by their nature but that can be regarded as multi-particle (for example, the Darboux-Halphen system and its generalizations). A wide class of two and three-particle polynomial dynamical systems is integrated.

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“…Suppose w n (z) is the solution of equation (8). Let w n (z) be given by the form (9). Expanding the polynomial w n (z) by the Taylor formula at the point z 0 .…”

Section: Lemmamentioning

confidence: 99%

Point vortices in the plane: positive-dimensional configurations

Demina

Kudryashov

Semenova

2017

J. Phys.: Conf. Ser.

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Abstract. The problem of constructing and classifying stationary equilibria of point vortices in the plane is studied. An ordinary differential equation that enables one to find positions of point vortices with circulations Γ1, Γ2, and Γ3 in stationary equilibrium is obtained. A necessary condition of an equilibrium existing is derived. The case of point vortex systems consisting of n + 2 point vortices with n vortices of circulation Γ 1 and two vortices of circulations Γ2 = aΓ1 and Γ3 = bΓ1, where a and b are integers, is considered in detail. The properties of polynomial solutions of the corresponding ordinary differential equation are investigated. A set of positivedimensional equilibrium configurations is found. A continuous free parameter is presented in the coefficients of corresponding polynomial solutions. These free parameters affect the positions of the roots and hence the vortex positions. Stationary equilibrium that could be derived from each other by rotation, extension, parallel translation is considered as equivalent. All found configurations seem to be new.

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Point vortices and classical orthogonal polynomials

Cited by 12 publications

References 23 publications

Electrostatic Partners and Zeros of Orthogonal and Multiple Orthogonal Polynomials

Electrostatic Partners and Zeros of Orthogonal and Multiple Orthogonal Polynomials

Multi-particle dynamical systems and polynomials

Point vortices in the plane: positive-dimensional configurations

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