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Regular and Chaotic Dynamics

, Volume 21, Issue 3, pp 249–253 | Cite as

Point vortex equilibria related to Bessel polynomials

  • Kevin A. O’NeilEmail author
Article

Abstract

The method of polynomials is used to construct two families of stationary point vortex configurations. The vortices are placed at the reciprocals of the zeroes of Bessel polynomials. Configurations that translate uniformly, and configurations that are completely stationary, are obtained in this way.

Keywords

point vortex equilibrium polynomial method 

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References

  1. 1.
    Aref, H., Beelen, P., and Brons, M., Bilinear Relative Equilibria of Identical Point Vortices, J. Nonlinear Sci., 2012, vol. 22, no. 5, pp. 849–885.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Aref, H., Newton, P.K., Stremler, M.A., Tokieda, T., and Vainchtein, D. L., Vortex Crystals, Adv. Appl. Mech., 2003, vol. 39, pp. 1–79.CrossRefGoogle Scholar
  3. 3.
    Clarkson, P., Vortices and Polynomials, Stud. Appl. Math., 2009, vol. 123, no. 1, pp. 37–62.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    de Bruin, M.G., Saff, E.B., and Varga, R. S., On the Zeros of Generalized Bessel Polynomials: 1, 2, Nederl. Akad. Wetensch. Indag. Math., 1981, vol. 43, no. 1, pp. 1–25.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    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. 562–576.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    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
  7. 7.
    Demina, M.V. and Kudryashov, N.A., Rotation, Collapse, and Scattering of Point Vortices, Theor. Comput. Fluid Dyn., 2014, vol. 28, no. 3, pp. 357–368.MathSciNetCrossRefGoogle Scholar
  8. 8.
    Grosswald, E., Bessel Polynomials, Lecture Notes in Math., vol. 698, New York: Springer, 1978.Google Scholar
  9. 9.
    O’Neil, K. A., Minimal Polynomial Systems for Point Vortex Equilibria, Phys. D, 2006, vol. 219, no. 1, pp. 69–79.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    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

Copyright information

© Pleiades Publishing, Ltd. 2016

Authors and Affiliations

  1. 1.Department of MathematicsThe University of TulsaTulsaUSA

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