Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 117, 14 pages      arXiv:1804.11273
Contribution to the Special Issue on Painlevé Equations and Applications in Memory of Andrei Kapaev

Truncated Solutions of Painlevé Equation ${\rm P}_{\rm V}$

Rodica D. Costin
The Ohio State University, 231 W 18th Ave, Columbus, OH 43210, USA

Received May 01, 2018, in final form October 25, 2018; Published online October 31, 2018

We obtain convergent representations (as Borel summed transseries) for the five one-parameter families of truncated solutions of the fifth Painlevé equation with nonzero parameters, valid in half planes, for large independent variable. We also find the position of the first array of poles, bordering the region of analyticity. For a special value of this parameter they represent tri-truncated solutions, analytic in almost the full complex plane, for large independent variable. A brief historical note, and references on truncated solutions of the other Painlevé equations are also included.

Key words: Painlevé trascendents; the fifth Painlevé equation; truncated solutions; poles of truncated solutions.

pdf (386 kb)   tex (23 kb)


  1. Ablowitz M.J., Clarkson P.A., Solitons, nonlinear evolution equations and inverse scattering, London Mathematical Society Lecture Note Series, Vol. 149, Cambridge University Press, Cambridge, 1991.
  2. Andreev F.V., Kitaev A.V., Exponentially small corrections to divergent asymptotic expansions of solutions of the fifth Painlevé equation, Math. Res. Lett. 4 (1997), 741-759.
  3. Boutroux P., Recherches sur les transcendantes de M. Painlevé et l'étude asymptotique des équations différentielles du second ordre (suite), Ann. Sci. École Norm. Sup. (3) 31 (1914), 99-159.
  4. Clarkson P.A., Painlevé equations - nonlinear special functions, in Orthogonal Polynomials and Special Functions, Lecture Notes in Math., Vol. 1883, Springer, Berlin, 2006, 331-411.
  5. Costin O., On Borel summation and Stokes phenomena for rank-$1$ nonlinear systems of ordinary differential equations, Duke Math. J. 93 (1998), 289-344, math.CA/0608408.
  6. Costin O., Asymptotics and Borel summability, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, Vol. 141, CRC Press, Boca Raton, FL, 2009.
  7. Costin O., Costin R.D., On the formation of singularities of solutions of nonlinear differential systems in antistokes directions, Invent. Math. 145 (2001), 425-485, math.CA/0202234.
  8. Costin O., Costin R.D., Asymptotic properties of a family of solutions of the Painlevé equation $\rm P_{VI}$, Int. Math. Res. Not. 2002 (2002), 1167-1182, math.CA/0202235.
  9. Costin O., Costin R.D., Huang M., Tronquée solutions of the Painlevé equation PI, Constr. Approx. 41 (2015), 467-494, arXiv:1310.5330.
  10. Costin O., Costin R.D., Huang M., A direct method to find Stokes multipliers in closed form for ${\rm P}_1$ and more general integrable systems, Trans. Amer. Math. Soc. 368 (2016), 7579-7621, arXiv:1205.0775.
  11. Costin O., Huang M., Tanveer S., Proof of the Dubrovin conjecture and analysis of the tritronquée solutions of ${\rm P}_{\rm I}$, Duke Math. J. 163 (2014), 665-704, arXiv:1209.1009.
  12. 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 (2009), 57-94, arXiv:0704.0501.
  13. Fokas A.S., Its A.R., Kapaev A.A., Novokshenov V.Yu., Painlevé transcendents: the Riemann-Hilbert approach, Mathematical Surveys and Monographs, Vol. 128, Amer. Math. Soc., Providence, RI, 2006.
  14. Grava T., Kapaev A., Klein C., On the tritronquée solutions of ${\rm P}_{\rm I}^2$, Constr. Approx. 41 (2015), 425-466, arXiv:1306.6161.
  15. Gray J.J., Fuchs and the theory of differential equations, Bull. Amer. Math. Soc. (N.S.) 10 (1984), 1-26.
  16. Gromak V.I., Laine I., Shimomura S., Painlevé differential equations in the complex plane, De Gruyter Studies in Mathematics, Vol. 28, Walter de Gruyter & Co., Berlin, 2002.
  17. Guzzetti D., A review of the sixth Painlevé equation, Constr. Approx. 41 (2015), 495-527, arXiv:1210.0311.
  18. Huang M., Xu S.-X., Zhang L., Location of poles for the Hastings-McLeod solution to the second Painlevé equation, Constr. Approx. 43 (2016), 463-494, arXiv:1410.3338.
  19. Its A.R., Kapaev A.A., Connection formulae for the fourth Painlevé transcendent; Clarkson-McLeod solution, J. Phys. A: Math. Gen. 31 (1998), 4073-4113.
  20. Its A.R., Kapaev A.A., Quasi-linear Stokes phenomenon for the second Painlevé transcendent, Nonlinearity 16 (2003), 363-386, nlin.SI/0108010.
  21. Jimbo M., Miwa T., Môri Y., Sato M., Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent, Phys. D 1 (1980), 80-158.
  22. Joshi N., Kitaev A.V., On Boutroux's tritronquée solutions of the first Painlevé equation, Stud. Appl. Math. 107 (2001), 253-291.
  23. Joshi N., Mazzocco M., Existence and uniqueness of tri-tronquée solutions of the second Painlevé hierarchy, Nonlinearity 16 (2003), 427-439, math.CA/0212117.
  24. Kapaev A.A., Asymptotic behavior of the solutions of the Painlevé equation of the first kind, Differential Equations 24 (1988), 1107-1115.
  25. Kapaev A.A., Global asymptotics of the fourth Painlevé transcendent, Steklov Math. Inst. and IUPUI, Preprint \# 96-06, 1996, available at
  26. Kapaev A.A., Quasi-linear Stokes phenomenon for the Painlevé first equation, J. Phys. A: Math. Gen. 37 (2004), 11149-11167, nlin.SI/0404026.
  27. Kapaev A.A., Quasi-linear Stokes phenomenon for the Hastings-McLeod solution of the second Painlevé equation, nlin.SI/0411009.
  28. Kapaev A.A., Kitaev A.V., Connection formulae for the first Painlevé transcendent in the complex domain, Lett. Math. Phys. 27 (1993), 243-252.
  29. Kowalevski S., Sur le problème de la rotation d'un corps solide autour d'un point fixe, in The Kowalevski Property (Leeds, 2000), CRM Proc. Lecture Notes, Vol. 32, Amer. Math. Soc., Providence, RI, 2002, 315-372, Reprinted from Acta Math. 12 (1889), 177-232.
  30. Lin Y., Dai D., Tibboel P., Existence and uniqueness of tronquée solutions of the third and fourth Painlevé equations, Nonlinearity 27 (2014), 171-186, arXiv:1306.1317.
  31. Novokshenov V.Yu., Distributions of poles to Painlevé transcendents via Padé approximations, Constr. Approx. 39 (2014), 85-99.
  32. Painlevé P., Oeuvres de Paul Painlevé, Tome III, Équations différentielles du second ordre, Mécanique, Quelques documents, Éditions du Centre National de la Recherche Scientifique, Paris, 1975.
  33. Parusnikova A., Asymptotic expansions of solutions to the fifth Painlevé equation in neighbourhoods of singular and nonsingular points of the equation, in Formal and Analytic Solutions of Differential and Difference Equations, Banach Center Publ., Vol. 97, Polish Acad. Sci. Inst. Math., Warsaw, 2012, 113-124.
  34. Shimomura S., Truncated solutions of the fifth Painlevé equation, Funkcial. Ekvac. 54 (2011), 451-471.
  35. Takei Y., On the connection formula for the first Painlevé equation - from the viewpoint of the exact WKB analysis, Surikaisekikenkyusho Kokyuroku 931 (1995), 70-99.
  36. Tracy C.A., Widom H., On exact solutions to the cylindrical Poisson-Boltzmann equation with applications to polyelectrolytes, Phys. A 244 (1997), 402-413, cond-mat/9701067.
  37. Wu T.T., McCoy B.M., Tracy C.A., Barouch E., Spin-spin correlation functions for the two-dimensional Ising model: exact theory in the scaling region, Phys. Rev. B 13 (1976), 316-374.
  38. Xia X., Tronquée solutions of the third and fourth Painlevé equations, SIGMA 14 (2018), 095, 28 pages, arXiv:1803.11230.

Previous article  Next article   Contents of Volume 14 (2018)