Regular and Chaotic Dynamics

, Volume 24, Issue 1, pp 90–100 | Cite as

Rational and Special Solutions for Some Painlevé Hierarchies

  • Nikolay A. KudryashovEmail author


A self-similar reduction of the Korteweg–de Vries hierarchy is considered. A linear system of equations associated with this hierarchy is presented. This Lax pair can be used to solve the Cauchy problem for the studied hierarchy. It is shown that special solutions of the self-similar reduction of the KdV hierarchy are determined via the transcendents of the first Painlevé hierarchy. Rational solutions are expressed by means of the Yablonskii–Vorob’ev polynomials. The map of the solutions for the second Painlevé hierarchy into the solutions for the self-similar reduction of the KdV hierarchy is illustrated using the Miura transformation. Lax pairs for equations of the hierarchy for the Yablonskii–Vorob’ev polynomial are discussed. Special solutions to the hierarchy for the Yablonskii–Vorob’ev polynomials are given. Some other hierarchies with properties of the Painlevé hierarchies are presented. The list of nonlinear differential equations whose general solutions are expressed in terms of nonclassical functions is extended.


self-similar reduction KdV hierarchy Painlevé hierarchy Painlevé transcendent transformation 

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  1. 1.
    Painlevé, P., Sur les équations différentielles du second ordre et d’ordre supérieur dont l’intégrale générale est uniforme, Acta Math., 1902, vol. 25, pp. 1–85.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Gambier, B., Sur les équations différentielles dont l’intégrate générale est uniforme, C. R. Acad. Sci. Paris, 1906, vol. 142, pp. 266–269, 1403–1406, 1497–1500.zbMATHGoogle Scholar
  3. 3.
    Borisov, A. V. and Kudryashov, N.A., Paul Painlevé and His Contribution to Science, Regul. Chaotic Dyn., 2014, vol. 19, no. 1, pp. 1–19.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Ablowitz, M. J. and Segur, H., Exact Linearization of a Painlevé Transcendent, Phys. Rev. Lett., 1977, vol. 38, no. 20, pp. 1103–1106.MathSciNetCrossRefGoogle Scholar
  5. 5.
    Ablowitz, M. J., Ramani, A., and Segur, H., A Connection between Nonlinear Evolution Equations and Ordinary Differential Equations of P-Type: 1, J. Math. Phys., 1980, vol. 21, no. 4, pp. 715–721.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Gromak, V. I., Painlevé Differential Equations in the Complex Plane, New York: de Gruyter, 2002.CrossRefzbMATHGoogle Scholar
  7. 7.
    Ablowitz, M. J. and Clarkson, P.A., Solitons, Nonlinear Evolution Equations and Inverse Scattering, London Math. Soc. Lect. Note Ser., vol. 149, Cambridge: Cambridge University Press, 1991.Google Scholar
  8. 8.
    Umemura, H., Second Proof of Irreducibility of the First Differential Equation by Painlevé, Nagoya Math. J., 1990, vol. 117, pp. 125–171.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kudryashov, N.A., The First and Second Painlevé Equations of Higher Order and Some Relations between Them, Phys. Lett. A, 1997, vol. 224, no. 6, pp. 353–360.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kudryashov, N.A., On New Transcendents Defined by Nonlinear Ordinary Differential Equations, J. Phys. A, 1998, vol. 31, no. 6, L129–L137.MathSciNetCrossRefGoogle Scholar
  11. 11.
    Kudryashov, N.A. and Pickering, A., Rational Solutions for Schwarzian Integrable Hierarchies, J. Phys. A, 1998, vol. 31, no. 47, pp. 9505–9518.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hone, A.N.W., Non-Autonomous Hénon–Heiles Systems, Phys. D, 1998, vol. 118, nos. 1–2, pp. 1–16.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Gordoa, P.R. and Pickering, A., Nonisospectral Scattering Problem: A Key to Integrable Hierachies, J. Math. Phys., 1999, vol. 40, no. 11, pp. 5749–5786.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kudryashov, N.A., Transcendents Defined by Nonlinear Fourth-Order Differential Equations, J. Phys. A, 1999, vol. 31, no. 6, pp. 999–1013.MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Mugan, U. and Jrad, F., Painlevé Test and the First Painlevé Hierarchy, J. Phys. A, 1999, vol. 32, no. 45, pp. 7933–7952.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kudryashov, N.A., Double Bäcklund Transformations and Special Integrals for the KII Hierarchy, Phys. Lett. A, 2000, vol. 273, no. 3, pp. 194–202.MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Pickering, A., Coalescence Limits for Higher Order Painlevé Equations, Phys. Lett. A, 2002, vol. 301, nos. 3–4, pp. 275–280.MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kudryashov, N.A., One Generalization of the Second Painlevé Hierarchy, J. Phys. A, 2002, vol. 35, no. 1, pp. 93–99.MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Kudryashov, N.A., Amalgamations of the Painlevé Equations, J. Math. Phys., 2003, vol. 44, no. 12, pp. 6160–6178.MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Kawai, T., Koike, T., Nishikawa, Y., and Takei, Y., On the Stokes Geometry of Higher Order Painlevé Equations, in Analyse complexe, systèmes dynamiques, sommabilité des séries divergentes et théories galoisiennes: 2, Astérisque, vol. 297, Paris: Soc. Math. France, 2004, pp. 117–166.Google Scholar
  21. 21.
    Shimomura, S., A Certain Expression of the First Painlevé Hierarchy, Proc. Japan Acad. Ser. A Math. Sci., 2004, vol. 80, no. 6, pp. 105–109.MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Shimomura, S., Poles and α-Poles of Meromorphic Solutions of the First Painlevé Hierarchy, Publ. Res. Inst. Math. Sci., 2004, vol. 40, no. 2, pp. 471–485.MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Gordoa, P.R., Joshi, N., and Pickering, A., Second and Fourth Painlevé Hierarchies and Jimbo–Miwa Linear Problems, J. Math. Phys., 2006, vol. 47, no. 7, 073504, 16 pp.Google Scholar
  24. 24.
    Mazzocco, M. and Mo, M.Y., The Hamiltonian Structure of the Second Painlevé Hierarchy, Nonlinearity, 2007, vol. 20, no. 12, pp. 2845–2882.MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Aoki, T., Multiple-Scale Analysis for Higher-Order Painlevé Equations, RIMS Kôkyûroke Bessatsu, 2008, B5, pp. 89–98.zbMATHGoogle Scholar
  26. 26.
    Kudryashov, N.A., Special Polynomials Associated with Some Hierarchies, Phys. Lett. A, 2008, vol. 372, no. 12, pp. 1945–1956.MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Kudryashov, N.A., Higher Painlevé Transcendents As Special Solutions of Some Nonlinear Integrable Hierarchies, Regul. Chaotic Dyn., 2014, vol. 19, no. 1, pp. 48–63.MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Gordoa, P.R. and Pickering, A., On a Extended Second Painlevé Hierarchy, J. Differential Equations, 2017, vol. 263, no. 7, pp. 4070–4125.MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Gordoa, P.R. and Pickering, A., Bäcklund Transformations for a New Extended Painlevé Hierarchy, Commun. Nonlinear Sci. Numer. Simul., 2019, vol. 69, pp. 78–97.Google Scholar
  30. 30.
    Kudryashov, N.A., Soliton, Rational and Special Solutions of the Korteweg–deVries Hierarchy, Appl. Math. Comput., 2010, vol. 217, no. 4, pp. 1774–1779.MathSciNetzbMATHGoogle Scholar
  31. 31.
    Kudryashov, N.A., Nonlinear Differential Equations Associated with the First Painlevé Hierarchy, Appl. Math. Lett., 2019, vol. 90, pp. 223–228.MathSciNetCrossRefGoogle Scholar
  32. 32.
    Polyanin, A.D. and Zaitsev, V. F., Handbook of Exact Solutions for Ordinary Differential Equations, 2nd ed., Boca Raton,Fla.: Chapman & Hall/CRC, 2002.CrossRefzbMATHGoogle Scholar
  33. 33.
    Yablonskii, A. I., On Rational Solutions of the Second Painlevé Equation, Vesti AN BSSR, Ser. Fiz.-Tech. Nauk, 1959, no. 3, pp. 30–35 (Russian).Google Scholar
  34. 34.
    Vorob’ev, A.P., On Rational Solutions of the Second Painlevé Equation, Differ. Uravn., 1965, vol. 1, pp. 79–81 (Russian).Google Scholar
  35. 35.
    Demina, M.V. and Kudryashov, N.A., The Generalized Yablonskii–Vorob’ev Polynomials and Their Properties, Phys. Lett. A, 2008, vol. 372, no. 29, pp. 4885–4890.MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Cole, J.D., On a Quasi-Linear Parabolic Equation occurring in Aerodynamics, Quart. Appl. Math., 1951, vol. 9, pp. 225–236.MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Hopf, E., The Partial Differential Equation ut + uux = μuxx, Comm. Pure Appl. Math., 1950, vol. 3, pp. 201–230.CrossRefGoogle Scholar
  38. 38.
    Kudryashov, N.A., From Singular Manifold Equations to Integrable Evolution Equations, J. Phys. A, 1994, vol. 27, no. 7, pp. 2457–2470.MathSciNetCrossRefzbMATHGoogle Scholar

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© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Department of Applied MathematicsNational Research Nuclear University MEPhIMoscowRussia

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