Sur les transformations de Schlesinger de la sixième équation de Painlevé

Abstract

Résumé. Après la récente découverte de deux nouvelles transformations de Schlesinger (TS) pour la sixièmeéquation de Painlevé, nous donnons les relations d'interdépendance de toutes les TS connues. Nous isolons ainsi l'unique d'entre elles quià la fois conserve la variable indépendante et n'est pas un produit d'autres TS.On the Schlesinger transformations of the sixth Painlevé equation Abstract. Following the recent discovery of two new Schlesinger transformations (ST) for the sixth Painlevé equation, we give the interrelations between all the known STs. We thus isolate the unique one which at the same time conserves the independent variable and is not a product of other STs.

A Truncation for Obtaining all the First Degree Birational Transformations of the Painlevé Transcendents

“…A method allowing one to derive such birational transformations was recently introduced [1,2], and later improved [3,4] so as to provide birational transformations which have a degree equal to one for any first degree N -th order ODE. Its application to the master Painlevé equation P6, P6 : u = 1 2 2 , provided the birational transformation [5], already found by Okamoto [6, p. 356],…”

“…Therefore the equation (3.4) still holds when, simultaneously, g 0 (U, x) vanishes and U satisfies the equation (3.3). In the case of P6, the corresponding valuesg 2 andg 0 are defined by the Riccati subequation, 5) and by the formula (2.5) applied tog 2 ,…”

On the Transformations of the Sixth Painlevé Equation

“…What the optimal Lax pair should be None of the above two new Lax pairs, with types (4, 0, 1) and (3,1,0), is yet optimal. To be optimal, a Lax pair should, in our opinion, obey the following criteria:…”

“…It is then easy to generate balanced expressions in which u 12 and u 21 become expressions which only differ by signs. Let us indeed denote T G the birational transformation of Garnier [8,9] (this is basically the square [1] of the elementary transformation [15]) and S a the operator which reverses the sign of θ ∞ . Then one has the diagram…”

New contiguity relation of the sixth Painlevé equation from a truncation

“…The transformation of Garnier is an integer power [12] of this first degree transformation. To achieve our goal (rely only on the singularity structure to find a birational transformation), we need to improve the singular manifold method so that it succeeds to obtain a birational transformation for P6.…”