Regular and Chaotic Dynamics

, Volume 23, Issue 5, pp 580–582 | Cite as

Relations Satisfied by Point Vortex Equilibria with Strength Ratio −2

  • Kevin A. O’NeilEmail author


Relations satisfied by the roots of the Loutsenko sequence of polynomials are derived. These roots are known to correspond to families of stationary and uniformly translating point vortices with two vortex strengths in ratio −2. The relations are analogous to those satisfied by the roots of the Adler–Moser polynomials, corresponding to equilibria with ratio −1. The proof uses an analysis of the differential equation that these polynomial pairs satisfy.


point vortex polynomial equilibrium 

MSC2010 numbers

76B47 37F10 34M15 


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  1. 1.
    Airault, H., McKean, H.P., and Moser, J., Rational and Elliptic Solutions of the Korteweg–deVries Equation and a Related Many-Body Problem, Comm. Pure Appl. Math., 1977, vol. 30, no. 1, pp. 95–148.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    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.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Aref, H., Newton, P.K., Stremler, M.A., Tokieda, T., and Vainchtein, D., Vortex Crystals, in Advances in Applied Mechanics: Vol. 39, E. van derGiessen, H. Aref (Eds.), San Diego: Acad. Press, 2003, pp. 1–79.Google Scholar
  4. 4.
    Aref, H., Vortices and Polynomials, Fluid Dynam. Res., 2007, vol. 39, nos. 1–3, pp. 5–23.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Burchnall, J. L. and Chaundy, T.W., A Set of Differential Equations Which Can Be Solved by Polynomials, Proc. London Math. Soc. (2), 1930, vol. 30, no. 6, pp. 401–414.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Clarkson, P.A., Vortices and Polynomials, Stud. Appl. Math., 2009, vol. 123, no. 1, pp. 37–62.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Kadtke, H.B. and Campbell, L. J., Method for Finding Stationary States of Point Vortices, Phys. Rev. A, 1987, vol. 36, no. 1, pp. 4360–4370.CrossRefGoogle Scholar
  8. 8.
    Kudryashov, N.A. and Demina, M.V., Relations between Zeros of Special Polynomials Associated with the Painlevé Equations, Phys. Lett. A, 2007, vol. 368, nos. 3–4, pp. 227–234.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kudryashov, N.A., Special Polynomials Associated with Some Hierarchies, Phys. Lett. A, 2008, vol. 372, no. 12, pp. 1945–1956.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    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, no. 19, 195205, 12 pp.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Loutsenko, I., Equilibrium of Charges and Differential Equations Solved by Polynomials, J. Phys. A, 2004, vol. 37, no. 4, pp. 1309–1321.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    O’Neil, K. A., Minimal Polynomial Systems for Point Vortex Equilibria, Phys. D, 2006, vol. 219, no. 1, pp. 69–79.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    O’Neil, K. A. and Cox-Steib, N., Generalized Adler–Moser and Loutsenko Polynomials for Point Vortex Equilibria, Regul. Chaotic Dyn., 2014, vol. 19, no. 5, pp. 523–532.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    O’Neil, K. A., Point Vortex Equilibria Related to Bessel Polynomials, Regul. Chaotic Dyn., 2016, vol. 21, no. 3, pp. 249–253.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Department of MathematicsThe University of TulsaTulsaUSA

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