The sixth Painlevé equation as similarity reduction of gl 3 generalized Drinfel'd-Sokolov hierarchy

Letters in Mathematical Physics

Volumn 79, Issue 3, 2007, Pages 221-234

  Kakei, Saburo ^a   Kikuchi, Tetsuya ^b  

^a RIKKYO UNIVERSITY   (Japan)

^b UNIVERSITY OF TOKYO   (Japan)

Author keywords

Drinfel'd Sokolov hierarchy; Painlev VI equation; Similarity reduction; Three wave resonant system

Indexed keywords

EID: 33947381014     PISSN: 03779017     EISSN: 15730530     Source Type: Journal    
DOI: 10.1007/s11005-007-0144-4     Document Type: Article

References (27)

1. Flaschka, H., Newell, A.C.: Monodromy and spectrum preserving deformations. I. Comm. Math. Phys. 76, 65-116 (1980)
2. Beals, R., Sattinger, D.H.: Integrable systems and isomonodromy deformations. Phys. D 65, 17-47 (1993)
3. Drinfel'd, V.G., Sokolov, V.V.: Lie algebras and equations of Korteweg - de Vries type. J Sov. Math. 30, 1975-2036 (1985)
4. de Groot, M.F., Hollowood, T.J., Miramontes, J.L.: Generalized Drinfel'd - Sokolov hierarchies, Commun. Math. Phys. 145, 57-84 (1992)
5. (1) . Funkcial. Ekvac. 41, 483-503 (1998)
6. Kakei, S., Kikuchi, T.: Affine Lie group approach to a derivative nonlinear Schrödinger equation and its similarity reduction. Int. Math. Res. Not. 78, 4181-4209 (2004)
7. Kakei, S., Kikuchi, T.: Solutions of a derivative nonlinear Schrödinger hierarchy and its similarity reduction. Glasgow Math. 47, Issue A , 99-107 (2005)
8. Kikuchi, T., Ikeda, T., Kakei, S.: Similarity reduction of the modified Yajima-Oikawa equation. J. Phys. A Math. Gen. 36, 11465-11480 (2003)
9. Fokas, A.S., Yortsos, Y.C.: The transformation properties of the sixth Painlevé equation and one-parameter families of solutions. Lett. Nuovo Cimento 30, 539-544 (1981)
10. Kitaev, A.V.: On similarity reductions of the three-wave resonant system to the Painlevé equations. J. Phys. A Math. Gen. 23, 3543-3553 (1990)
11. Conte, R., Grundland, A.M., Musette, M.: A reduction of the resonant three-wave interaction to the generic sixth Painlevé equation. J. Phys. A Math. Gen. 39, 12115-12127 (2006)
12. Jimbo, M., Miwa, T.: Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II. Physica D 2, 407-448 (1981)
13. Okamoto, K.: Isomonodromic deformation and Painlevé equations, and the Gamier system. J. Fac. Sci. Univ. Tokyo, Sect.IA Math. 33, 575-618 (1986)
14. Harnad, J.: Dual isomonodromic deformations and moment maps to loop algebras. Commun. Math. Phys. 166, 337-365 (1994)
15. Mazzocco, M.: Painlevé sixth equation as isomonodromic deformations equation of an irregular system. In:Kuznetsov V.B.(ed.), The Kowalevski property (CRM Proceedings and Lecture Notes 32), pp. 219-238 (2002)
16. Burroughs, N.J., Groot, M.F. de , Hollowood, T..T., Miramontes, J.L.: Generalized Drinfeld-Sokolov hierarchies II: the Hamiltonian structures. Commun. Math. Phys. 153, 187-215 (1993)
17. Wilson, G.: The τ-functions of the gAKNS equations, In: Babelon, O., Cartier, P., Kosmann-Schwarzbach, Y.(eds.) Verdier Memorial Conference on Integrable Systems pp. 131-145. Birkhauser, Basel(1993)
18. Bergvelt, M.J., ten Kroode, A.P.E.: Partitions, vertex operator constructions and multi-component KP equations. Pac. J Math. 171, 23-88 (1995)
19. Kac, V.G.: Infinite Dimensional Lie Algebras, 3rd edn., Cambridge University Press, Cambridge, MA (1990)
20. Bergvelt, M.J., ten Kroode, A.P.E.: τ functions and zero curvature equations of Toda-AKNS type. J. Math. Phys. 29, 1308-1320 (1988)
21. Ueno, K., Takasaki, K.: Toda lattice hierarchy. Adv. Stud. Pure Math. 4, 1-95 (1984)
22. Kac, V.G., van de Leur, J.: The n-component KP hierarchy and representation theory, In:Fokas, A.S., Zakharov, V.E.(eds.) Important Developments in Soliton Theory, pp. 302-343 Springer, Berlin (1993)
23. Dubrovin, B.: Geometry of 2d topological field theories, In: Francaviglia, M., Greco, S.(eds.) Integrable Systems and Quantum Groups (Springer Lecture Notes in Math. 1620). pp. 120-348. Springer, Berlin (1996)
24. Aratyn, H., Leur, J. van de : Integrable structure behind WDVV equations. Theor. Math. Phys. 134, 14-26 (2003)
25. Okamoto, K.: Studies on the Painlevé equations I Sixth Painlevé equation PVI. Ann. Mat. Pura Appl. 146(4), 337-381 (1987)
26. Mazzocco, M.: Irregular isomonodromic deformations for Gamier systems and Okamoto's canonical transformations. J. Lond. Math. Soc. 70, 405-419 (2004)
27. Boalch, P.: From Klein to Painlevé via Fourier, Laplace and Jimbo. Proc. Lond. Math. Soc. 90, 167-208 (2005)