Solution Hierarchies for the Painlevé IV Equation

  • David Bermúdez
  • David J. Fernández C.
Conference paper
Part of the Trends in Mathematics book series (TM)


We will obtain real and complex solutions of the Painlevé IV equation through supersymmetric quantum mechanics. The real solutions will be classified into several hierarchies, and a similar procedure will be followed for the complex solutions.


Factorization method supersymmetric quantum mechanics Painlevé equations 


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  1. 1.
    A.P. Veselov, A.B. Shabat. Dressing chains and spectral theory of the Schrödinger operator. Funct. Anal. Appl. 27 (1993) 81–96.MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    V.E. Adler. Nonlinear chains and Painlevé equations. Physica D 73 (1994) 335–351.MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    K. Iwasaki, H. Kimura, S. Shimomura, M. Yoshida. From Gauss to Painlevé: a modern theory of special functions, Aspects of Mathematics E 16 Vieweg, Braunschweig, Germany, 1991.Google Scholar
  4. 4.
    M.J. Ablowitz, P.A. Clarkson. Solitons, nonlinear evolution equations and inverse scattering, Cambridge University Press, New York, 1992.Google Scholar
  5. 5.
    P. Winternitz. Physical applications of Painlevé type equations quadratic in highest derivative. Painlevé trascendents, their asymptotics and physical applications, NATO ASI Series B, New York (1992) 425–431.Google Scholar
  6. 6.
    V.E. Matveev, M.A. Salle. Darboux transformation and solitons, Springer, Berlin, 1991.Google Scholar
  7. 7.
    L. Infeld, T. Hull. The factorization method. Rev. Mod. Phys. 23 (1951) 21–68.MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    B. Mielnik, Factorization method and new potentials with the oscillator spectrum. J. Math. Phys. 25 (1984) 3387–3389.MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    D. Bermúdez, D.J. Fernández. Supersymmetric quantum mechanics and Painlevé I V equation. SIGMA 7 (2011) 025, 14 pages.Google Scholar
  10. 10.
    D. Bermúdez, D.J. Fernández. Non-hermitian Hamiltonians and the Painlevé I V equation with real parameters. Phys. Lett. A 375 (2011) 2974–2978.MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    A.A. Andrianov, M. Ioffe, V. Spiridonov. Higher-derivative supersymmetry and the Witten index. Phys. Lett. A 174 (1993) 273–279.MathSciNetCrossRefGoogle Scholar
  12. 12.
    B. Mielnik, O. Rosas-Ortiz. Factorization: little or great algorithm? J. Phys. A: Math. Gen. 37 (2004) 10007–10035.MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    D.J. Fernández. Supersymmetric quantum mechanics. AIP Conf. Proc. 1287 (2010) 3–36.Google Scholar
  14. 14.
    D.J. Fernández, V. Hussin. Higher-order SUSY, linearized nonlinear Heisenberg algebras and coherent states. J. Phys. A: Math. Gen. 32 (1999) 3603–3619.CrossRefMATHGoogle Scholar
  15. 15.
    D.J. Fernández, J. Neg ro, L.M. Nieto. Elementary systems with partial finite ladder spectra. Phys. Lett. A 324 (2004) 139–144.Google Scholar
  16. 16.
    J.M. Carballo, D.J. Fernández, J. Neg ro, L.M. Nieto. Polynomial Heisenberg algebras. J. Phys. A: Math. Gen. 37 (2004) 10349–10362.Google Scholar
  17. 17.
    A.A. Andrianov, F. Cannata, M.V. Ioffe, D. Nishnianidze. Systems with higher-order shape invariance: spectral and algebraic properties, Phys. Lett. A 266 (2000) 341–349.MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    A.A. Andrianov, F. Cannata, J.P. Dedonder, M.V. Ioffe. SUSY quantum mechanics with complex superpotentials and real energy spectra. Int. J. Mod. Phys. A 14 (1999) 2675–2688.MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    G. Junker, P. Roy. Conditionally exactly solvable potentials: a supersymmetric construction method. Ann. Phys. 270 (1998) 155–164.MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    A.P. Bassom, P.A. Clarkson, A.C. Hicks. Bäcklund transformations and solution hierarchies for the fourth Painlevé equation. Stud. Appl. Math. 95 (1995) 1–75.MathSciNetMATHGoogle Scholar
  21. 21.
    C. Rogers, W.F. Shadwick. Bäcklund transformations and their applications, Academic Press, London, 1982.Google Scholar

Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.Departamento de FísicaCinvestavMéxico D.F.Mexico

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