Archive for Rational Mechanics and Analysis

, Volume 202, Issue 3, pp 707–785 | Cite as

Okamoto’s Space for the First Painlevé Equation in Boutroux Coordinates



We study the completeness and connectedness of asymptotic behaviours of solutions of the first Painlevé equation d2 y/dx 2 = 6 y 2 + x in the limit \({x\to\infty,x\in{\mathbb C}}\). This problem arises in various physical contexts including the critical behaviour near gradient catastrophe for the focusing nonlinear Schrödinger equation. We prove that the complex limit set of solutions is non-empty, compact and invariant under the flow of the limiting autonomous Hamiltonian system, that the infinity set of the vector field is a repellor for the dynamics and obtain new proofs for solutions near the equilibrium points of the autonomous flow. The results rely on a realization of Okamoto’s space, that is, the space of initial values compactified and regularized by embedding in \({{\mathbb C}{\mathbb P} 2}\) through an explicit construction of nine blowups.


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  1. 1.
    Anosov, D.V.: Geodesic Flows on Closed Riemann Manifolds with Negative Curvature. English translation of the Proceedings of the Steklov Institute of Mathematics, Vol. 90, 1967. American Mathematical Society, Providence, Rhode Island, 1969Google Scholar
  2. 2.
    Boutroux, P.: Recherches sur les transcendantes de M. Painlevé et l’étude asymptotique des équations différentielles du second ordre. Ann. Sci. École Norm. Sup., Série 3 30, 255–375 (1913). (suite) (3) 31, 99–159 (1914)Google Scholar
  3. 3.
    Bruns H.: Ueber die Perioden der elliptischen Integrale erster und zweiter Gattung. Festschrift, Dorpat, 1875(Reprinted in Mathematische Annalen 27), 234–252 (1886)MathSciNetGoogle Scholar
  4. 4.
    Coddington E.A., Levinson N.: Theory of Ordinary Differential Equations. McGraw-Hill, New York (1955)MATHGoogle Scholar
  5. 5.
    Costin O.: On Borel summation and Stokes phenomena for rank one nonlinear systems of ODE’s. Duke Math. J. 93, 289–344 (1998)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Costin O.: Correlation between pole location and asymptotic behavior for Painlevé I solutions. Commun. Pure Appl. Math. 52, 461–478 (1999)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Costin O., Costin R.D.: On the formation of singularities of solutions of nonlinear differential systems in antistokes directions. Invent. Math. 145, 425–485 (2001)MathSciNetADSMATHCrossRefGoogle Scholar
  8. 8.
    Cotton E.: Sur les solutions asymptotiques des équations différentielles. Ann. Sci. École Norm. Sup., Série 3(28), 473–521 (1911)MathSciNetGoogle Scholar
  9. 9.
    Dubrovin B., Grava T., Klein C.: On universality of critical behavior in the focusing nonlinear Schrödinger equation, elliptic umbilic catastrophe and the tritronquée solution to the Painlevé-I equation. J. Nonlinear Sci. 19, 57–94 (2009)MathSciNetADSMATHCrossRefGoogle Scholar
  10. 10.
    Duistermaat, J.J.: Okamoto’s space for the Painlevé equation P I. Preprint, May 2009Google Scholar
  11. 11.
    Gérard et R., Levelt A.H.M.: Sur les connexions à singularités régulières dans le cas de plusieurs variables. Funkcialaj Ekvacioj 19, 149–173 (1976)MathSciNetGoogle Scholar
  12. 12.
    Hadamard J.: Sur l’itération et les solutions asymptotiques des équations différentielles. Bull. Soc. Math. France 29, 224–228 (1901)MATHGoogle Scholar
  13. 13.
    Holmes P., Spence D.: On a Painlevé-type boundary value problem. Q. J. Mech. Appl. Math. 37, 525–538 (1984)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Joshi N., Kitaev A.V.: On Boutroux’s tritronquée solutions of the first Painlevé equation. Stud. Appl. Math. 107, 253–291 (2001)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Joshi N., Kruskal M.D.: An asymptotic approach to the connection problem for the first and the second Painlevé equations. Phys. Lett. A 130, 129–137 (1988)MathSciNetADSCrossRefGoogle Scholar
  16. 16.
    Joshi N., Kruskal M.D.: The Painlevé connection problem: an asymptotic approach. I. Stud. Appl. Math. 86, 315–376 (1992)MathSciNetMATHGoogle Scholar
  17. 17.
    Kajiwara K., Masuda T., Noumi M., Ohta Y., Yamada Y.: Cubic pencils and Painlevé Hamiltonians. Funkcialaj Ekvacioj 48, 147–160 (2005)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Kitaev A.V.: Elliptic asymptotics of the first and second Painlevé transcendents. Uspekhi Mat. Nauk 49, 77–140 (1994)MathSciNetGoogle Scholar
  19. 19.
    Kapaev A.A.: Quasi-linear Stokes phenomenon for the Painlevé first equation. J. Phys. A 37, 11149–11167 (2004)MathSciNetADSMATHCrossRefGoogle Scholar
  20. 20.
    Malgrange B.: Sur le théorème de Maillet. Asymptot. Anal. 2, 1–4 (1989)MathSciNetMATHGoogle Scholar
  21. 21.
    Okamoto K.: Sur les feuilletages associés aux équation du second ordre à points critiques fixes de P. Painlevé. Espaces de conditions initiales. Jpn. J. Math. 5, 1–79 (1979)MATHGoogle Scholar
  22. 22.
    Perron O.: Über die Stabilität und asymptotisches Verhalten der Integrale von Differentialgleichungensystemen. Math. Z. 29, 129–160 (1928)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Persson U.: Configurations of Kodaira fibers on rational elliptic surfaces. Math. Z. 205, 1–47 (1990)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Schmickler-Hirzebruch, U.: Elliptische Flächen über \({{\mathbb P} _1({\mathbb C})}\) mit drei Ausnahmefasern und die hypergeometrische Differentialgleichung. Schriftenreihe des Mathematischen Instituts der Universität Münster, No. 33 (1985)Google Scholar
  25. 25.
    Shioda T., Takano K.: On some Hamiltonian structures of Painlevé systems, I. Funkcialaj Ekvacioj 40, 271–291 (1997)MathSciNetMATHGoogle Scholar
  26. 26.
    Stokes, G.G.: On the discontinuity of arbitrary constants which appear in divergent developments. Trans. Camb. Phil. Soc. 10, 106–128 (1857). In Mathematical and Physical Papers by late Sir George Gabriel Stokes, Vol. 4, pp. 77–109. Cambridge University Press (1904)Google Scholar

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© Springer-Verlag 2011

Authors and Affiliations

  1. 1.School of Mathematics and Statistics F07The University of SydneySydneyAustralia

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