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Communications in Mathematical Physics

, Volume 242, Issue 1–2, pp 251–275 | Cite as

Integrable Dynamics of Charges Related to the Bilinear Hypergeometric Equation

  • Igor LoutsenkoEmail author
Article

Abstract

A family of systems related to a linear and bilinear evolution of roots of polynomials in the complex plane is introduced. Restricted to the line, the evolution induces dynamics of the Coulomb charges (or point vortices) in external potentials, while its fixed points correspond to equilibriums of charges in the plane. The construction reveals a direct connection with the theories of the Calogero-Moser systems and Lie-algebraic differential operators. A study of the equilibrium configurations amounts in a construction (bilinear hypergeometric equation) for which the classical orthogonal and the Adler-Moser polynomials represent some particular cases.

Keywords

Vortex Differential Operator Complex Plane Direct Connection External Potential 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Adler, M., Moser, J.: On a class of polynomials connected with the Korteveg-de Vries equation. Comm. Math. Phys. 61, 1–30 (1978)Google Scholar
  2. 2.
    Aref, H.: Integrable, Chaotic and turbulent motion of vortices in two dimensional flows. Ann. Rew. Fluid Mech. 15, 345–389 (1983)CrossRefzbMATHGoogle Scholar
  3. 3.
    Arnold, V.I., Khesin, B.A.: Topological Methods in Hydrodynamics. NY: Springer-Verlag, 1998Google Scholar
  4. 4.
    Bartman, A.B.: A new interpretation of the Adler-Moser KdV polynomials: interaction of vortices. Nonlinear and Turbulent Processes in Physics, Vol. 3 (Kiev 1983), Chur: Harwood Academic Publ., 1984, pp. 1175–1181Google Scholar
  5. 5.
    Berest, Y.: Huygens principle and the bispectarl problem. CRM Proceedings and Lecture Notes 14, 1998Google Scholar
  6. 6.
    Berest, Y., Loutsenko, I.: Huygens principle in Minkowski space and soliton solutions of the Korteveg de-Vries equation. Commun. Math. Phys. 190, 113–132 (1997)CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    Burchnall, J.L., Chaundy, T.W.: A set of differential equations which can be solved by polynomials. Proc. London Math. Soc. 30, 401–414 (1929)zbMATHGoogle Scholar
  8. 8.
    Calogero, F.: Motion of poles and zeros of special solutions of nonlinear and linear partial differential equations and related ‘‘solvable’’ many body problems. Nuovo Cimento 43 B, 177 (1978)MathSciNetGoogle Scholar
  9. 9.
    Chalych, O., Feigin, M., Veselov, A.: Multidimensional Baker-Akhieser Functions and Huygens Principle. Commun. Math. Phys. 206, 533–566 (1999)CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    Choodnovsky, D., Choodnovsky, G.: Pole expansions of nonlinear partial differential equations. Nuovo Cimento 40 B, 339 (1977)Google Scholar
  11. 11.
    Crum, M.: Associated Sturm-Liouville Systems. Quart. J. Math 6, 121–127 (1955)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Gurevich, B.B.: Foundations of the Theory of Algebraic Invariants. Groningen, Holland: P. Noordhoff, 1964Google Scholar
  13. 13.
    Hadamard, J.: Lectures on Cauchy’s Problem in Linear Partial Differential Equations. New Haven, CT: Yale Univ. Press, 1923Google Scholar
  14. 14.
    Heckman, G.J., Opdam, E.M.: Root systems and hypergeometric functions I. Composito Math. 64 329–352 (1987)Google Scholar
  15. 15.
    Inozemtsev, V.: On the motion of classical integrable systems of interacting particles in an external field. Phys. Lett. 98, 316–318 (1984)Google Scholar
  16. 16.
    Krichever, I.: Methods of algebraic geometry in the theory of nonlinear equations. Russ. Math. Surv. 32, 185–213 (1977)zbMATHGoogle Scholar
  17. 17.
    Krichever, I.: Rational solutions of the Kadomtsev-Petviashvilli equation and integrable systems of N particles on line. Funct. Anal. Appl. 12, 76–78 (1978)zbMATHGoogle Scholar
  18. 18.
    Novikov, S., Pitaevski, L., Zakharov, V., Manakov, S.: Theory of Solitons: Inverse Scattering Method. New York, NY: Contemporary Soviet Mathematics, 1984Google Scholar
  19. 19.
    Perelomov, A.M.: Integrable Systems in Classical Mechanics and Lie’s Algebras. Moscow: Nauka 1990Google Scholar
  20. 20.
    Szegö, G.: Orthogonal Polynomials. NY: AMS, 1939Google Scholar
  21. 21.
    Turbiner, A.V.: Quasi-exactly solvable problems and sl(2) algebra. Commun. Math. Phys. 118, 467 (1988)zbMATHGoogle Scholar
  22. 22.
    Veselov, A.: Rational solutions of the Kadomtsev-Petviashvilli equation and Hamiltonian systems. Russ. Math. Surv. 35, 239–240 (1980)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  1. 1.SISSATriesteItaly

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