logo ANNALES DE L'INSTITUT FOURIER

Avec cedram.org

Table des matières de ce fascicule | Article précédent | Article suivant
Henrik Aratyn; Johan van de LEUR
The symplectic Kadomtsev-Petviashvili hierarchy and rational solutions of Painlevé VI
(La hiérarchie de Kadomtsev-Petviashvili symplectique et la solution ratio\-nnelle de Painlevé VI)
Annales de l'institut Fourier, 55 no. 6 (2005), p. 1871-1903, doi: 10.5802/aif.2145
Article PDF | Analyses MR 2187939 | Zbl 1093.14015
Class. Math.: 14M15, 17B65, 17B80, 22E67, 34M55, 37K10, 37K35
Mots clés: hiérarchie de Kadomtsev-Petviashvili, formulation Grassmanienne, variétes de Frobenius, déformation isomonodromique, painlevé VI

Résumé - Abstract

Nous établissons des connexions entre une certaine classe d' équations de Painlevé VI paramétrée par une dimension conforme $\mu$, des équations de type Euler top dépendant du temps, des déformations et des variétés de Frobenius de dimensions 3. Nous construisons explicitement la fonction isomonodromique tau et des solutions d'équations de type Euler top en terme de solutions wronskiennes de la hiérarchie de Kadomtsev-Petviashvili symplectique à 1 contrainte et 2 vecteurs. Nous utilisons ici la formulation grasmannienne. Ces solutions wronskiennes donnent des solutions rationelles de l'équations de Painlevé VI pour $\mu=1,2,{\ldots} $

Bibliographie

[1] H. Aratyn, J.F. Gomes, J.W. van de Leur & A.H. Zimerman, “WDVV equations, Darboux-Egoroff metric and the dressing method”, contribution to the UNESP2002 workshop on Integrable Theories, Solitons and Duality, http://jhep.sissa.it or [arXiv:math-ph/0210038, 2002 arXiv |  MR 2170155 |  Zbl 1096.53054
[2] H. Aratyn & J. van de Leur, Integrable structures behind WDVV equations, Teor. Math. Phys., 2003, p. 14-26  Zbl 02194695
[3] H. Aratyn & J. van de Leur, “Solutions of the WDVV Equations and Integrable Hierarchies of KP Type”, Commun. Math. Phys. 239 (2003), p. 155-182 Article |  MR 1997119 |  Zbl 01969276
[4] H. Aratyn, E. Nissimov & S. Pacheva, Multi-component matrix KP hierarchies as symmetry-enhanced scalar KP hierarchies and their Darboux-Bäcklund solutions, in Bäcklund and Darboux transformations., CRM Proc. Lecture Notes, Amer. Math. Soc., 2001, p. 109-120  Zbl 0999.37041
[5] E. Date, M. Jimbo, M. Kashiwara & T. Miwa, “Transformation groups for soliton equations. 6. KP hierarchies of orthogonal and symplectic type”, J. Phys. Soc., Japan 50 (1981), p. 3813-3818 Article |  MR 638808 |  Zbl 0571.35102
[6] B. Dubrovin, Integrable systems and classification of 2-dimensional topological field theories, Birkhäuser, 1993, p. 313-359  Zbl 0824.58029
[7] B. Dubrovin, Geometry on 2D topological field theories, Lecture Notes in Math., Springer Berlin, 1996, p. 120-348  Zbl 0841.58065
[8] B. Dubrovin & M. Mazzocco, “Monodromy of certain Painlevé VI trascendents and reflection groups”, Invent. Math. 141 (2000), p. 55-147 Article |  MR 1767271 |  Zbl 0960.34075
[9] B. Dubrovin & Y.J. Zhang, “Frobenius manifolds and Virasoro constraints.”, Selecta Math. (N.S.) 5 (1999) no. 4, p. 423-466 Article |  MR 1740678 |  Zbl 0963.81066
[10] G. F. Helminck & J. W. van de Leur, “Geometric Bäcklund-Darboux transformations for the KP hierarchy”, Publ. Res. Inst. Math. Sci. 37 (2001) no. 4, p. 479-519 Article |  MR 1865402 |  Zbl 1028.37043
[11] G. F. Helminck & J. W. van de Leur, “An analytic description of the vector constrained KP hierarchy”, Commun. Math Phys. 193 (1998), p. 627-641 Article |  MR 1624847 |  Zbl 0907.35115
[12] G. F. Helminck & J. W. van de Leur, Constrained and Rational Reductions of the KP hierarchy, Springer Lecture Notes in Physics, 1998, p. 167-182  Zbl 0901.35088
[13] N. J. Hitchin, “Twistor spaces, Einstein metrics and isomonodromic deformations”, J. Diff. Geom. 42 (1995), p. 30-112  MR 1350695 |  Zbl 0861.53049
[14] N. J. Hitchin, Poncelet polygons and the Painlevé transcendents, Geometry and Analysis, Oxford University Press, 1996, p. 151-185  Zbl 0893.32018
[15] N. J. Hitchin, A new family of Einstein metrics, manifolds and geometry, Manifolds and geometry (Pisa, 1993), Sympos. Math., XXXVI, Cambridge Univ. Press, 1996, p. 190-222  Zbl 0858.53038
[16] M. Jimbo, T. Miwa & K. Ueno, “Monodromy preserving deformations of linear ordinary differential equations with rational coefficients I”, Physica 2D 2 (1981), p. 306-352 Article |  MR 630674
[17] M. Jimbo & T. Miwa, “Monodromy preserving deformations of linear ordinary differential equations with rational coefficients II”, Physica 2D 3 (1981), p. 407-448  MR 625446
[18] M. Jimbo & T. Miwa, “Monodromy preserving deformations of linear ordinary differential equations with rational coefficients III”, Physica 2D 4 (1981), p. 26-46 Article |  MR 636469
[19] M. Jimbo, M. & T. Miwa, “Solitons and Infinite Dimensional Lie Algebras”, Publ. RIMS, Kyoto Univ. 19 (1983), p. 943-1001 Article |  MR 723457 |  Zbl 0557.35091
[20] V.G. Kac & J.W. van de Leur, “The $n$-component $KP$ hierarchy and representation theory, Integrability, topological solitons and beyond”, J. Math. Phys. 44 (2003) no. 8, p. 3245-3293  MR 2006751 |  Zbl 1062.37071
[21] J.W. van de Leur, “Twisted $GL_n$ Loop Group Orbit and Solutions of WDVV Equations”, Internat. Math. Res. Notices 11 (2001), p. 551-573  MR 1836730 |  Zbl 0991.37042
[22] J.W. van de Leur & R. Martini, “The construction of Frobenius Manifolds from KP tau-Functions”, Commun. Math. Phys. 205 (1999), p. 587-616 Article |  MR 1711265 |  Zbl 0940.53046
[23] I.G. Macdonald, Symmetric functions and Hall polynomials. Second edition., Oxford Mathematical Monographs, Oxford University Press, New York, 1995  MR 1354144 |  Zbl 0824.05059
[24] G. Mahoux, Introduction to the theory of isomonodromic deformations of linear ordinary differential equations with rational coefficients, The Painlevé property, one century later, CRM series in mathematical physics, Springer, 1999, p. 35-76  Zbl 1034.34105
[25] M. Mazzocco, Picard and Chazy solutions to the Painlevé VI equation, Math. Annalen, 2001, p. 131-169  Zbl 0999.34079
[26] T. Shiota, Prym varieties and soliton equations, Adv. Ser. Math. Phys., World Sci. Publishing, 1989, p. 407-448  Zbl 0766.14020
haut