Skip to main content
SpringerLink
Log in

A Computational Exploration of the Second Painlevé Equation

  • Published:
Foundations of Computational Mathematics Aims and scope Submit manuscript

Abstract

The pole field solver developed recently by the authors (J. Comput. Phys. 230:5957–5973, 2011) is used to survey the space of solutions of the second Painlevé equation that are real on the real axis. This includes well-known solutions such as the Hastings–McLeod and Ablowitz–Segur type of solutions, as well as some novel solutions. The speed and robustness of this pole field solver enable the exploration of pole dynamics in the complex plane as the parameter and initial conditions of the differential equation are varied. Theoretical connection formulas are also verified numerically.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25

Similar content being viewed by others

References

  1. M.J. Ablowitz, H. Segur, Exact linearization of a Painlevé transcendent, Phys. Rev. Lett. 38, 1103–1106 (1977).

    Article  MathSciNet  Google Scholar 

  2. M.J. Ablowitz, H. Segur, Solitons and the inverse scattering transform (SIAM, Philadelphia, 1981).

    Book  MATH  Google Scholar 

  3. A.P. Bassom, P.A. Clarkson, C.K. Law, J.B. McLeod, Application of uniform asymptotics to the second Painlevé transcendent, Arch. Ration. Mech. Anal. 143, 241–271 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  4. M. Bertola, On the location of poles for the Ablowitz–Segur family of solutions of the second Painlevé equation, Nonlinearity 25, 1179–1185 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  5. P. Boutroux, Remarques sur les singularités transcendantes des fonctions de deux variables, Bull. Soc. Math. Fr. 39, 296–304 (1911).

    MathSciNet  MATH  Google Scholar 

  6. P. Boutroux, Recherches sur les transcendantes de M. Painlevé et l’étude asymptotique des équations différentielles du second ordre (suite), Ann. Sci. École Norm. Super. (3) 31, 99–159 (1914).

    MathSciNet  MATH  Google Scholar 

  7. T. Claeys, A.B.J. Kuijlaars, M. Vanlessen, Multi-critical unitary random matrix ensembles and the general Painlevé II equation, Ann. Math. (2) 168, 601–641 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  8. P.A. Clarkson, Painlevé equations—nonlinear special functions, in Orthogonal polynomials and special functions. Lecture notes in mathematics, vol. 1883 (Springer, Berlin, 2006), pp. 331–411.

    Chapter  Google Scholar 

  9. P.A. Clarkson, Asymptotics of the second Painlevé equation, in Special functions and orthogonal polynomials. Contemporary mathematics, vol. 471 (Am. Math. Soc., Providence, 2008), pp. 69–83.

    Chapter  Google Scholar 

  10. P.A. Clarkson, Painlevé transcendents, in NIST handbook of mathematical functions (U.S. Dept. Commerce, Washington, 2010), pp. 723–740.

    Google Scholar 

  11. P.A. Clarkson, E.L. Mansfield, The second Painlevé equation, its hierarchy and associated special polynomials, Nonlinearity 16, R1–R26 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  12. P.A. Clarkson, J.B. McLeod, A connection formula for the second Painlevé transcendent, Arch. Ration. Mech. Anal. 103, 97–138 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  13. A.S. Fokas, A.R. Its, A.A. Kapaev, V.Y. Novokshenov, Painlevé transcendents: the Riemann–Hilbert approach (Am. Math. Soc., Providence, 2006).

    Book  Google Scholar 

  14. B. Fornberg, J.A.C. Weideman, A numerical methodology for the Painlevé equations, J. Comput. Phys. 230, 5957–5973 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  15. V.I. Gromak, Solutions of the second Painlevé equation, Differ. Uravn. 18, 753–763, 914–915 (1982).

    MathSciNet  Google Scholar 

  16. V.I. Gromak, I. Laine, S. Shimomura, Painlevé differential equations in the complex plane (de Gruyter, Berlin, 2002).

    Book  MATH  Google Scholar 

  17. S.P. Hastings, J.B. McLeod, A boundary value problem associated with the second Painlevé transcendent and the Korteweg–de Vries equation, Arch. Ration. Mech. Anal. 73, 31–51 (1980).

    Article  MathSciNet  MATH  Google Scholar 

  18. A.R. Its, A.A. Kapaev, Quasi-linear Stokes phenomenon for the second Painlevé transcendent, Nonlinearity 16, 363–386 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  19. A.A. Kapaev, Private communication.

  20. A.A. Kapaev, Global asymptotics of the second Painlevé transcendent, Phys. Lett. A 167, 356–362 (1992).

    Article  MathSciNet  Google Scholar 

  21. A.V. Kashevarov, The second Painlevé equation in electrostatic probe theory: numerical solutions, Comput. Math. Math. Phys. 38, 950–958 (1998).

    MathSciNet  MATH  Google Scholar 

  22. A.V. Kashevarov, The second Painlevé equation in the electrostatic probe theory: numerical solutions for the partial absorption of charged particles by the surface, Tech. Phys. 49, 3–9 (2004).

    Article  Google Scholar 

  23. A.V. Kitaev, Symmetric solutions for the first and the second Painlevé equation, J. Math. Sci. 73, 494–499 (1995).

    Article  MathSciNet  Google Scholar 

  24. B.M. McCoy, S. Tang, Connection formulae for Painlevé functions, Physica D 18, 190–196 (1986).

    Article  MathSciNet  MATH  Google Scholar 

  25. J.W. Miles, On the second Painlevé transcendent, Proc. R. Soc. Lond. Ser. A 361, 277–291 (1978).

    Article  MathSciNet  MATH  Google Scholar 

  26. V.Y. Novokshenov, Padé approximations for Painlevé I and II transcendents, Theor. Math. Phys. 159, 853–862 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  27. S. Olver, Numerical solution of Riemann–Hilbert problems: Painlevé II, Found. Comput. Math. 11, 153–179 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  28. P. Painlevé, Mémoire sur les équations différentielles dont l’intégrale générale est uniforme, Bull. Soc. Math. Fr. 28, 201–261 (1900).

    MATH  Google Scholar 

  29. P. Painlevé, Sur les équations différentielles du second ordre et d’ordre supérieur dont l’intégrale générale est uniforme, Acta Math. 25, 1–85 (1902).

    Article  MathSciNet  Google Scholar 

  30. J.A. Reeger, B. Fornberg, Painlevé IV with both parameters zero: a numerical study, Stud. Appl. Math. 130, 108–133 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  31. R.R. Rosales, The similarity solution for the Korteweg–de Vries equation and the related Painlevé transcendent, Proc. R. Soc. Lond. Ser. A 361, 265–275 (1978).

    Article  MathSciNet  MATH  Google Scholar 

  32. H. Segur, M.J. Ablowitz, Asymptotic solutions of nonlinear evolution equations and a Painlevé transcendent, Physica D 3, 165–184 (1981).

    Article  MATH  Google Scholar 

  33. C.A. Tracy, H. Widom, Painlevé functions in statistical physics, Publ. Res. Inst. Math. Sci. 47, 361–374 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  34. J.A.C. Weideman, dip.sun.ac.za/~weideman/PAINLEVE/.

  35. I.M. Willers, A new integration algorithm for ordinary differential equations based on continued fraction approximations, Commun. ACM 17, 504–508 (1974).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

Financial support for this work was provided by NSF grant DMS-0914647 (first author) and the National Research Foundation in South Africa (second author), as well as the National Institute for Theoretical Physics (NITheP) in South Africa. Communications with Andrew Bassom, Peter Clarkson, Percy Deift, Alexander Its, Andrei Kapaev, Jonah Reeger and Harvey Segur are also acknowledged. The workshop “Numerical solution of the Painlevé Equations”, held in May 2010 at the International Center for the Mathematical Sciences (ICMS), in Edinburgh, stimulated [14] and the present study.

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, University of Colorado, 526 UCB, Boulder, CO, 80309, USA

    Bengt Fornberg

  2. National Institute for Theoretical Physics (NITheP), Western Cape, South Africa

    Bengt Fornberg

  3. Department of Mathematical Sciences, University of Stellenbosch, Private Bag X1, Matieland, 7602, South Africa

    J. A. C. Weideman

Authors
  1. Bengt Fornberg
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. A. C. Weideman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bengt Fornberg.

Additional information

Communicated by Elizabeth Mansfield.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fornberg, B., Weideman, J.A.C. A Computational Exploration of the Second Painlevé Equation. Found Comput Math 14, 985–1016 (2014). https://doi.org/10.1007/s10208-013-9156-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10208-013-9156-x

Keywords

Mathematics Subject Classification (2010)

Search

Navigation