Advertisement

Computational Methods and Function Theory

, Volume 3, Issue 1, pp 117–126 | Cite as

Airy Solutions of Painlevé’s Second Equation

  • Norbert Steinmetz
Article

Abstract

We prove that any transcendental solution of Painlevé’s second equation w″ = α + zw + 2w 3, which has the form w = R(z,u), with R rational in both variables and non-linear with respect to u, is obtained by repeated application of the Bäcklund transformation to some solution of the Riccati equation U′ = ±(z/2 + U 2). In particular, \(\alpha = n+1/2, n\in {\rm Z}\), and w has order of growth ϱ 3/2. Moreover it is shown that u satisfies some Riccati differential equation u’ = a(z) + b(z)u + c(z)u with rational coefficients.

Keywords

Painlevé Airy and Riccati differential equation Bäcklund transformation 

2000 MSC

34M55 30D35 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    L. Bieberbach, Theorie der gewöhnlichen Differentialgleichungen, Springer, Berlin, 1965.MATHCrossRefGoogle Scholar
  2. 2.
    V. I. Gromak, One-parameter systems of solutions of Painlevé’s equation; Engl. Transl.: Differential Equations 14 (1978), 1510–1513.MathSciNetMATHGoogle Scholar
  3. 3.
    V. I. Gromak, I. Laine, and S. Shimomura, Painlevé Differential Equations in the Complex Plane, Walter de Gruyter, Berlin, 2002.MATHCrossRefGoogle Scholar
  4. 4.
    W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford, 1975.Google Scholar
  5. 5.
    A. Hinkkanen and I. Laine, Solutions of the first and second Painlevé equations are meromorphic, Journal dAnalyse Math. 79 (1999), 345–377.MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    S. Shimomura, The first, the second and the fourth Painlevé transcendents are of finite order, Proc. Japan Acad. Ser. A 77 (2001), 42–45.MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    S. Shimomura, Growth of the first, the second and the fourth Painlevé transcendents, Math. Proc. Cambr. Philos. Soc 134 (2003), 259–269.MathSciNetMATHGoogle Scholar
  8. 8.
    —, Lower estimates for the growth of Painlevé transcendents, to appear in Funkcial. Ecvac.Google Scholar
  9. 9.
    N. Steinmetz, On Painlevé’s equations I, II and IV, J. Anal. Math. 82 (2000), 363–377.MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    N. Steinmetz, Value distribution of the Painlevé transcendents, Isr. J. Math. 128 (2002), 29–52.MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    —, Global properties of the Painlevé transcendents, preprint 2003, 23 p.Google Scholar
  12. 12.
    H. Wittich, Zur Theorie der Riccatischen Differentialgleichung, Math. Ann. 127 (1954), 433–440.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Heldermann  Verlag 2003

Authors and Affiliations

  1. 1.Fachbereich MathematikUniversität DortmundDortmundGermany

Personalised recommendations