The symplectic Kadomtsev-Petviashvili hierarchy and rational solutions of Painlevé VI
Annales de l'Institut Fourier, Volume 55 (2005) no. 6, pp. 1871-1903.

Equivalence is established between a special class of Painlevé VI equations parametrized by a conformal dimension μ, time dependent Euler top equations, isomonodromic deformations and three-dimensional Frobenius manifolds. The isomonodromic tau function and solutions of the Euler top equations are explicitly constructed in terms of Wronskian solutions of the 2-vector 1-constrained symplectic Kadomtsev-Petviashvili (CKP) hierarchy by means of Grassmannian formulation. These Wronskian solutions give rational solutions to the Painlevé VI equation for μ=1,2,...

Nous établissons des connexions entre une certaine classe d’ équations de Painlevé VI paramétrée par une dimension conforme μ, 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 μ=1,2,...

DOI: 10.5802/aif.2145
Classification: 14M15, 17B65, 17B80, 22E67, 34M55, 37K10, 37K35
Keywords: KP hierarchy, Grassmanian, Frobenius manifold, isomonodromic deformation, painlevé VI
Mot clés : hiérarchie de Kadomtsev-Petviashvili, formulation Grassmanienne, variétes de Frobenius, déformation isomonodromique, painlevé VI
Aratyn, Henrik 1; van de LEUR, Johan 

1 University of Illinois at Chicago, department of physics, 845 W. Taylor St., Chicago IL 60607-7059 (USA), University of Utrecht, Mathematical Institute, P.O. Box 80010, 3508 TA Utrecht (The Netherlands)
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Aratyn, Henrik; van de LEUR, Johan. The symplectic Kadomtsev-Petviashvili hierarchy and rational solutions of Painlevé VI. Annales de l'Institut Fourier, Volume 55 (2005) no. 6, pp. 1871-1903. doi : 10.5802/aif.2145. https://aif.centre-mersenne.org/articles/10.5802/aif.2145/

[1] H. Aratyn; J.F. Gomes; J.W. van de Leur; A.H. Zimerman WDVV equations, Darboux-Egoroff metric and the dressing method (2002) (contribution to the UNESP2002 workshop on Integrable Theories, Solitons and Duality, http://jhep.sissa.it or [arXiv:math-ph/0210038) | MR | Zbl

[2] H. Aratyn; J. van de Leur Integrable structures behind WDVV equations (Teor. Math. Phys.), Volume 134 (2003), pp. 14-26 | Zbl

[3] H. Aratyn; J. van de Leur Solutions of the WDVV Equations and Integrable Hierarchies of KP Type, Commun. Math. Phys., Volume 239 (2003), pp. 155-182 | DOI | MR | Zbl

[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., The geometry of solitons (Halifax, NS, 1999) (CRM Proc. Lecture Notes), Volume 29 (2001), pp. 109-120 | Zbl

[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, Volume 50 (1981), pp. 3813-3818 | DOI | MR | Zbl

[6] B. Dubrovin Integrable systems and classification of 2-dimensional topological field theories, Integrable Systems, proceedings of Luminy 1991 conference dedicated to the memory of J.-L. Verdier, eds. O. Babelon, P. Cartier, Y. Kosmann-Schwarzbach (1993), pp. 313-359 | Zbl

[7] B. Dubrovin Geometry on 2D topological field theories, Integrable Systems and Quantum Groups (Montecatini Terme, 1993) (Lecture Notes in Math.), Volume 1620 (1996), pp. 120-348 | Zbl

[8] B. Dubrovin; M. Mazzocco Monodromy of certain Painlevé VI trascendents and reflection groups, Invent. Math., Volume 141 (2000), pp. 55-147 | DOI | MR | Zbl

[9] B. Dubrovin; Y.J. Zhang Frobenius manifolds and Virasoro constraints., Selecta Math. (N.S.), Volume 5 (1999) no. 4, pp. 423-466 | DOI | MR | Zbl

[10] G. F. Helminck; J. W. van de Leur Geometric Bäcklund-Darboux transformations for the KP hierarchy, Publ. Res. Inst. Math. Sci., Volume 37 (2001) no. 4, pp. 479-519 | DOI | MR | Zbl

[11] G. F. Helminck; J. W. van de Leur An analytic description of the vector constrained KP hierarchy, Commun. Math Phys., Volume 193 (1998), pp. 627-641 | DOI | MR | Zbl

[12] G. F. Helminck; J. W. van de Leur; H. Aratyn. T.D. Imbo, W.-Y. Keung Constrained and Rational Reductions of the KP hierarchy, Supersymmetry and Integrable Models (Springer Lecture Notes in Physics), Volume 502 (1998), pp. 167-182 | Zbl

[13] N. J. Hitchin Twistor spaces, Einstein metrics and isomonodromic deformations, J. Diff. Geom., Volume 42 (1995), pp. 30-112 | MR | Zbl

[14] N. J. Hitchin Poncelet polygons and the Painlevé transcendents (Geometry and Analysis) (1996), pp. 151-185 | Zbl

[15] N. J. Hitchin A new family of Einstein metrics, manifolds and geometry (Manifolds and geometry (Pisa, 1993), Sympos. Math., XXXVI) (1996), pp. 190-222 | Zbl

[16] M. Jimbo; T. Miwa; K. Ueno Monodromy preserving deformations of linear ordinary differential equations with rational coefficients I, Physica 2D, Volume 2 (1981), pp. 306-352 | DOI | MR

[17] M. Jimbo; T. Miwa Monodromy preserving deformations of linear ordinary differential equations with rational coefficients II, Physica 2D, Volume 3 (1981), pp. 407-448 | MR

[18] M. Jimbo; T. Miwa Monodromy preserving deformations of linear ordinary differential equations with rational coefficients III, Physica 2D, Volume 4 (1981), pp. 26-46 | DOI | MR

[19] M. Jimbo; M.; T. Miwa Solitons and Infinite Dimensional Lie Algebras, Publ. RIMS, Kyoto Univ., Volume 19 (1983), pp. 943-1001 | DOI | MR | Zbl

[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., Volume 44 (2003) no. 8, pp. 3245-3293 | MR | Zbl

[21] J.W. van de Leur Twisted GL n Loop Group Orbit and Solutions of WDVV Equations, Internat. Math. Res. Notices, Volume 11 (2001), pp. 551-573 | MR | Zbl

[22] J.W. van de Leur; R. Martini The construction of Frobenius Manifolds from KP tau-Functions, Commun. Math. Phys., Volume 205 (1999), pp. 587-616 | DOI | MR | Zbl

[23] I.G. Macdonald Symmetric functions and Hall polynomials. Second edition., Oxford Mathematical Monographs, Oxford University Press, New York, 1995 | MR | Zbl

[24] G. Mahoux; R. Conte 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) (1999), pp. 35-76 | Zbl

[25] M. Mazzocco Picard and Chazy solutions to the Painlevé VI equation (Math. Annalen), Volume 321 (2001), pp. 131-169 | Zbl

[26] T. Shiota Prym varieties and soliton equations, Infinite-dimensional Lie algebras and groups (Luminy-Marseille, 1988) (Adv. Ser. Math. Phys.), Volume 7 (1989), pp. 407-448 | Zbl

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