Advertisement

Regular and Chaotic Dynamics

, Volume 17, Issue 5, pp 371–384 | Cite as

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 

MSC2010 numbers

33D45+76M23 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

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.MathSciNetCrossRefGoogle Scholar
  2. 2.
    Borisov, A.V. and Mamaev, I. S., Mathematical Methods of Dynamics of Vortex Structures, Moscow-Izhevsk: R&C Dynamics, ICS, 2005 (Russian).zbMATHGoogle Scholar
  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.CrossRefGoogle Scholar
  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.MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Aref, H., Vortices and Polynomials, Fluid Dynam. Res., 2007, vol. 39, nos. 1–3, pp. 5–23.MathSciNetzbMATHCrossRefGoogle Scholar
  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.CrossRefGoogle Scholar
  9. 9.
    O’Neil, K. A., Symmetric Configurations of Vortices, Phys. Lett. A, 1987, vol. 124, no. 9, pp. 503–507.MathSciNetCrossRefGoogle Scholar
  10. 10.
    O’Neil, K. A., Minimal Polynomial Systems for Point Vortex Equilibria, Phys. D, 2006, vol. 219, pp. 69–79.MathSciNetzbMATHCrossRefGoogle Scholar
  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.MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    O’Neil, K. A., Clustered Equilibria of Point Vortices, Regul. Chaotic Dyn., 2011, vol. 16, no. 6, pp. 555–561.MathSciNetCrossRefGoogle Scholar
  13. 13.
    Clarkson, P.A., Vortices and Polynomials, Stud. Appl. Math., 2009, vol. 123, no. 1, pp. 37–62.MathSciNetzbMATHCrossRefGoogle Scholar
  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.MathSciNetCrossRefGoogle Scholar
  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.MathSciNetzbMATHCrossRefGoogle Scholar
  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].MathSciNetCrossRefGoogle Scholar
  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.MathSciNetzbMATHCrossRefGoogle Scholar
  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.zbMATHGoogle Scholar
  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].MathSciNetCrossRefGoogle Scholar
  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.MathSciNetzbMATHGoogle Scholar
  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.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2012

Authors and Affiliations

  1. 1.Department of Applied MathematicsNational Research Nuclear University “MEPhI”MoscowRussian Federation

Personalised recommendations