A global factorization theorem for the ZS-AKNS system.
A Lax formalism for the elliptic difference Painlevé equation.
A particular solution of a Painlevé system in terms of the hypergeometric function .
A simple proof of the non-integrability of the first and the second Painlevé equations
The first and the second Painlevé equations are explicitly Hamiltonian with time dependent Hamilton function. By a natural extension of the phase space one gets corresponding autonomous Hamiltonian systems in ℂ⁴. We prove that the latter systems do not have any additional algebraic first integral. In the proof equations in variations with respect to a parameter are used.
An ultradiscrete matrix version of the fourth Painlevé equation.
Application of uniform asymptotics method to analyzing the asymptotic behaviour of the general fourth Painlevé transcendent.
Asymptotic expansions of solutions to the fifth Painlevé equation in neighbourhoods of singular and nonsingular points of the equation
Applying methods of plane Power Geometry we are looking for the asymptotic expansions of solutions to the fifth Painlevé equation in the neighbourhood of its singular and nonsingular points.
Bäcklund transformations for first and second Painlevé hierarchies.
Bessel's differential equation and its Hyers-Ulam stability.
Birational canonical transformations and classical solutions of the sixth Painlevé equation
Boutroux's method vs. re-scaling. Lower estimates for the orders of growth of the second and fourth Painlevé transcendents.
Darboux-Lamé equation and isomonodromic deformation.
Déformations isomonodromiques, forme de Liouville, fonction
Cet article améliore des résultats antérieurs de Miwa et de l’auteur sur la “fonction ” de l’équation de Schlesinger. On relie cette fonction à la forme de Liouville d’un groupe de lacets associé naturellement à cette équation
Dynamics on Character Varieties and Malgrange irreducibility of Painlevé VI equation
We consider representations of the fundamental group of the four punctured sphere into . The moduli space of representations modulo conjugacy is the character variety. The Mapping Class Group of the punctured sphere acts on this space by symplectic polynomial automorphisms. This dynamical system can be interpreted as the monodromy of the Painlevé VI equation. Infinite bounded orbits are characterized: they come from -representations. We prove the absence of invariant affine structure (and invariant...
Families of linear differential equations related to the second Painlevé equation
This paper is a sequel to [vdP-Sa] and [vdP]. The two classes of differential modules (0,-,3/2) and (-,-,3), related to PII, are interpreted as fine moduli spaces. It is shown that these moduli spaces coincide with the Okamoto-Painlevé spaces for the given parameters. The geometry of the moduli spaces leads to a proof of the Painlevé property for PII in standard form and in the Flaschka-Newell form. The Bäcklund transformations, the rational solutions and the Riccati solutions for PII are derived...
From holonomy of the Ising model form factors to -fold integrals and the theory of elliptic curves.
Generalization of Okamoto's equation to arbitrary Schlesinger system.
Global properties of the Painlevé transcendents: new results and open questions.
Groupe de Galois différentiel local et représentation adjointe
Dans cet article on s’intéresse à la représentation adjointe du tore exponentiel sur l’algèbre de Lie du groupe de Galois différentiel local. Nous proposons un algorithme pour réduire les sous-espaces poids de dimension supérieure à 1 à des sous-espaces de racines. Ce faisant, on construit un tore (en général) maximal qui contient le tore exponentiel. Au cours de ce travail on est amené à étudier la régularité du tore exponentiel dans le groupe de Galois local.
Hamiltonian structure of pi hierarchy.