Vortices and Polynomials

Abstract

The relationship between point vortex dynamics and the properties of polynomials with roots at the vortex positions is discussed. Classical polynomials, such as the Hermite polynomials, have roots that describe the equilibria of identical vortices on the line. Stationary and uniformly translating vortex configurations with vortices of the same strength but positive or negative orientation are given by the zeros of the Adler-Moser polynomials, which arise in the description of rational solutions of the Korteweg-de Vries equation. For quadupole background flow, vortex configurations are given by the zeros of polynomials expressed as wronskians of Hermite polynomials. Further new solutions are found in this case using the special polynomials arising the in the description of rational solutions of the fourth Painleve equation.

17 pages, minor revisions and references added

References in corpus (7)

• Application of the $\tau$-Function Theory of Painlevé Equations to Random Matrices: PIV, PII and the GUE
• Replica field theories, Painleve transcendents, and exact correlation functions
• Painleve IV and degenerate Gaussian Unitary Ensembles
• On Stieltjes relations, Painlevé-IV hierarchy and complex monodromy
• Equilibrium of charges and differential equations solved by polynomials
• Integrable Dynamics of Charges Related to Bilinear Hypergeometric Equation
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Cited by in corpus (17)

• Generating Functions, Polynomials and Vortices with Alternating Signs in Bose-Einstein Condensates
• Large-degree asymptotics of rational Painleve-II functions. II
• Connection between quantum systems involving the fourth Painleve transcendent and $k$-step rational extensions of the harmonic oscillator related to Hermite EOP
• Vortices ans Polynomials: Nonuniqueness of the Adler-Moser polynomials for the Tkachenko equation
• Point vortices and polynomials of the Sawada-Kotera and Kaup-Kupershmidt equations
• Shape invariance and equivalence relations for pseudowronskians of Laguerre and Jacobi polynomials
• Approximate Conservation Laws in the KdV Equation
• Point vortices and classical orthogonal polynomials
• A Fuchsian matrix differential equation for Selberg correlation integrals
• Rotating Equilibria of Vortex Sheets
• On Integrable Ermakov-Painlevé IV Systems
• Electrostatic partners and zeros of orthogonal and multiple orthogonal polynomials
• Exceptional orthogonal polynomials and generalized Schur polynomials
• Finiteness of Fixed Equilibrium Configurations of Point Vortices in the Plane with Background Flow
• Helicoids and vortices
• The correlation coefficient for vortices in the plane
• Equilibrium of Charges and Differential Equations Solved by Polynomials II