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

SIGMA 5 (2009), 024, 11 pages      arXiv:0903.0342      https://doi.org/10.3842/SIGMA.2009.024

Bäcklund Transformations for First and Second Painlevé Hierarchies

Ayman Hashem Sakka
Department of Mathematics, Islamic University of Gaza, P.O. Box 108, Rimal, Gaza, Palestine

Received November 25, 2008, in final form February 24, 2009; Published online March 02, 2009

Abstract
We give Bäcklund transformations for first and second Painlevé hierarchies. These Bäcklund transformations are generalization of known Bäcklund transformations of the first and second Painlevé equations and they relate the considered hierarchies to new hierarchies of Painlevé-type equations.

Key words: Painlevé hierarchies; Bäcklund transformations.

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References

  1. Airault H., Rational solutions of Painlevé equations, Stud. Appl. Math. 61 (1979), 31-53.
  2. Kudryashov N.A., The first and second Painlevé equations of higher order and some relations between them, Phys. Lett. A 224 (1997), 353-360.
  3. Hone A.N.W., Non-autonomous Hénon-Heiles systems, Phys. D 118 (1998), 1-16, solv-int/9703005.
  4. Kudryashov N.A., Soukharev M.B., Uniformization and transcendence of solutions for the first and second Painlevé hierarchies, Phys. Lett. A 237 (1998), 206-216.
  5. Kudryashov N.A., Fourth-order analogies of the Painlevé equations, J. Phys. A: Math. Gen. 35 (2002), 4617-4632.
  6. Kudryashov N.A., Amalgamations of the Painlevé equations, J. Math. Phys. 44 (2003), 6160-6178.
  7. Mugan U., Jrad F., Painlevé test and the first Painlevé hierarchy, J. Phys. A: Math. Gen. 32 (1999), 7933-7952.
  8. Mugan U., Jrad F., Painlevé test and higher order differntial equations, J. Nonlinear Math. Phys. 9 (2002), 282-310, nlin.SI/0301043.
  9. Gordoa P.R., Pickering A., Nonisospectral scattering problems: a key to integrable hierarchies, J. Math. Phys. 40 (1999), 5749-5786.
  10. Gordoa P.R., Joshi N., Pickering A., On a generalized 2+1 dispersive water wave hierarchy, Publ. Res. Inst. Math. Sci. 37 (2001), 327-347.
  11. Sakka A., Linear problems and hierarchies of Painlevé equations, J. Phys. A: Math. Theor. 42 (2009), 025210, 19 pages.
  12. Fokas A.S., Ablowitz M.J., On a unified approach to transformations and elementary solutions of Painlevé equations, J. Math. Phys. 23 (1982), 2033-2042.
  13. Okamoto K., Studies on the Painlevé equations. III. Second and fourth Painlevé equations, PII and PIV, Math. Ann. 275 (1986), 221-255.
  14. Gordoa P. R., Joshi N., Pickering A., Mappings preserving locations of movable poles: a new extension of the truncation method to ordinary differential equations, Nonlinearity 12 (1999), 955-968, solv-int/9904023.
  15. Gromak V., Laine I., Shimomura S., Painlevé differential equations in the complex plane, de Gruyter Studies in Mathematics, Vol. 28, Walter de Gruyter & Co., Berlin, 2002.
  16. Clarkson P.A., Joshi N., Pickering A., Bäcklund transformations for the second Pianlevé hierarchy: a modified truncation approach, Inverse Problems 15 (1999), 175-187, solv-int/9811014.
  17. Cosgrove C.M., Scoufis G., Painlevé classification of a class of differential equations of the second order and second degree, Stud. Appl. Math. 88 (1993), 25-87.
  18. Mugan U., Sakka A., Second-order second-degree Painlevé equations related with Painlevé I-VI equations and Fuchsian-type transformations, J. Math. Phys. 40 (1999), 3569-3587.
  19. Sakka A., Elshamy S., Bäcklund transformations for fourth-order Painlevé-type equations, The Islamic University Journal, Natural Science Series 14 (2006), 105-120.
  20. Cosgrove C. M., Higher-order Painlevé equations in the polynomial class. I. Bureau symbol P2, Stud. Appl. Math. 104 (2000), 1-65.
  21. Cosgrove C.M., Higher-order Painlevé equations in the polynomial class. II. Bureau symbol P1, Stud. Appl. Math. 116 (2006), 321-413.
  22. Sakka A., Bäcklund transformations for fifth-order Painlevé equations, Z. Naturforsch. A 60 (2005), 681-686.
  23. Sakka A., Second-order fourth-degree Painlevé-type equations, J. Phys. A: Math. Gen. 34 (2001), 623-631.

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