The symplectic Kadomtsev-Petviashvili hierarchy and rational solutions of Painlevé VI

Aratyn, Henrik; van de LEUR, Johan

Annales de l'Institut Fourier, Tome 55 (2005) no. 6, pp. 1871-1903.

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 $\mu =1,2,...$

Equivalence is established between a special class of Painlevé VI equations parametrized by a conformal dimension $\mu$, 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 $\mu =1,2,...$

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

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