Regular and Chaotic Dynamics

, Volume 17, Issue 5, pp 371–384

# Point vortices and classical orthogonal polynomials

• Maria V. Demina
• Nikolai A. Kudryashov
Article

## 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.

## Keywords

point vortices special polynomials classical orthogonal polynomials

33D45+76M23

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## References

1. 1.
Aref, H., Integrable, Chaotic, and Turbulent Vortex Motion in Two-Dimentional Flows, Annu. Rev. Fluid Mech., 1983, vol. 15, pp. 345–89.
2. 2.
Borisov, A.V. and Mamaev, I. S., Mathematical Methods of Dynamics of Vortex Structures, Moscow-Izhevsk: R&C Dynamics, ICS, 2005 (Russian).
3. 3.
Kadtke, H.B. and Campbell, L. J., Method for Finding Stationary States of Point Vortices, Phys. Rev. A, 1987, vol. 36, pp. 4360–4370.
4. 4.
Aref, H., Relative Equilibria of Point Vortices and the Fundamental Theorem of Algebra, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 2011, vol. 467, no. 2132, pp. 2168–2184.
5. 5.
Aref, H., Vortices and Polynomials, Fluid Dynam. Res., 2007, vol. 39, nos. 1–3, pp. 5–23.
6. 6.
Aref, H., Point Vortex Dynamics: A Classical Mathematics Playground, J. Math. Phys., 2007, vol. 48, no. 6, 065401, 23 pp.Google Scholar
7. 7.
Dirksen, T. and Aref, H., Close Pairs of Relative Equilibria for Identical Point Vortices, Phys. Fluids, 2011, vol. 23, no. 5, 051706, 4 pp.Google Scholar
8. 8.
Aref, H., Newton, P.K., Stremler, M.A., Tokieda, T., and Vainchtein, D., Vortex Crystals, Adv. Appl. Mech., 2003, vol. 39, pp. 1–79.
9. 9.
O’Neil, K. A., Symmetric Configurations of Vortices, Phys. Lett. A, 1987, vol. 124, no. 9, pp. 503–507.
10. 10.
O’Neil, K. A., Minimal Polynomial Systems for Point Vortex Equilibria, Phys. D, 2006, vol. 219, pp. 69–79.
11. 11.
O’Neil, K. A., Relative Equilibrium and Collapse Configurations of Four Point Vortices, Regul. Chaotic Dyn., 2007, vol. 12, no. 2, pp. 117–126.
12. 12.
O’Neil, K. A., Clustered Equilibria of Point Vortices, Regul. Chaotic Dyn., 2011, vol. 16, no. 6, pp. 555–561.
13. 13.
Clarkson, P.A., Vortices and Polynomials, Stud. Appl. Math., 2009, vol. 123, no. 1, pp. 37–62.
14. 14.
Demina, M.V. and Kudryashov, N.A., Point Vortices and Polynomials of the Sawada-Kotera and Kaup-Kupershmidt Equations, Regul. Chaotic Dyn., 2011, vol. 16, no. 6, pp. 56–576.
15. 15.
Demina, M.V. and Kudryashov, N.A., Vortices and Polynomials: Non-Uniqueness of the Adler-Moser Polynomials for the Tkachenko Equation, J. Phys. A, 2012, vol. 45,195205, 12 pp.Google Scholar
16. 16.
Adler, M. and Moser, J., On a Class of Polynomials Connected with the Korteweg-deVries Equation, Comm. Math. Phys., 1978, vol. 61, no. 1, pp. 1–30.
17. 17.
Bartman, A. B., A New Interpretation of the Adler-Moser KdV Polynomials: Interaction of Vortices, in Nonlinear and Turbulent Processes in Physics (Kiev, 1983): Vol. 3, R. Z. Sagdeev (Ed.), New York: Harwood Acad., 1984, pp. 1175–1181.Google Scholar
18. 18.
Demina, M.V. and Kudryashov, N.A., Special Polynomials and Rational Solutions of the Hierarchy of the Second Painlevé Equation, Teoret. Mat. Fiz., 2007, vol. 153, no. 1, pp. 58–67 [Theoret. and Math. Phys., 2007, vol. 153, no. 1, pp. 1398–1406].
19. 19.
Kudryashov, N.A. and Demina, M.V., The Generalized Yablonskii-Vorob’ev Polynomials and Their Properties, Phys. Lett. A, 2008, vol. 372, no. 29, pp. 4885–4890.
20. 20.
Tkachenko, V. K. Thesis, Institute of Physical Problems, Moscow, 1964.Google Scholar
21. 21.
Matveev, V.B. and Salle, M.A., Darboux Transformations and Solitons, Berlin: Springer, 1991.
22. 22.
Oblomkov, A.A., Monodromy-Free Schrödinger Operators with Quadratically Increasing Potentials, Teoret. Mat. Fiz., 1999, vol. 121, no. 3, pp. 374–386 [Theoret. and Math. Phys., 1999, vol. 121, no. 3, pp. 1574–1584].
23. 23.
Loutsenko, I., Integrable Dynamics of Charges Related to Bilinear Hypergeometric Equation, Comm. Math. Phys., 2003, vol. 242, nos. 1–2, pp. 251–275.
24. 24.
Filipuk, G.V. and Clarkson, P.A., The Symmetric Fourth Painlevé Hierarchy and Associated Special Polynomials, Stud. Appl. Math., 2008, vol. 121, no. 2, pp. 157–188.