Related papers: On Transformations in the Painlev\'e Family

In this paper some open problems for Painlev\'e equations are discussed. In particular the following open problems are described: (i) the Painlev\'e equivalence problem; (ii) notation for solutions of the Painlev\'e equations; (iii)…

This short review is an introduction to a great variety of methods, the collection of which is called the Painlev\'e analysis, intended at producing all kinds of exact (as opposed to perturbative) results on nonlinear equations, whether…

We present a systematic method for the construction of discrete Painlev\'e equations. The method, dubbed `restoration', allows one to obtain all discrete Painlev\'e equations that share a common autonomous limit, up to homographic…

For certain finite groups $G$ of B\"acklund transformations we show that the dynamics of $G$-invariant configurations of $n|G|$ Calogero--Painlev\'e particles is equivalent to certain $n$-particle Calogero--Painlev\'e system. We also show…

The group reduction procedure is applied to vector generalizations of the NLS, mKdV, and KdV equations. The resulting ODE systems admit isomonodromic Lax representations and are multicomponent generalizations of the Painlev\'e equations…

A new integrable sixth-order nonlinear wave equation is discovered by means of the Painleve analysis, which is equivalent to the Korteweg - de Vries equation with a source. A Lax representation and a Backlund self-transformation are found…

Discrete Painlev\'e equations constitute a famous class of integrable non-autonomous second order difference equations. A classification scheme proposed by Sakai interprets a discrete Painlev\'e equation as a birational map between…

We study the inverse problem in the theory of (standard) orthogonal polynomials involving two polynomials families $(P_n)_n$ and $(Q_n)_n$ which are connected by a linear algebraic structure such as $$P_n(x)+\sum_{i=1}^N…

The ``Painlev\'e analysis'' is quite often perceived as a collection of tricks reserved to experts. The aim of this course is to demonstrate the contrary and to unveil the simplicity and the beauty of a subject which is in fact the theory…

We show that there exists a rational change of coordinates of Painlev\'e's P1 equation $y''=6y^2+x$ and of the elliptic equation $y''=6y^2$ after which these two equations become analytically equivalent in a region in the complex phase…

We introduce certain B\"acklund transformations for rational solutions of the Painlev\'e VI equation. These transformations act ona family of Painlev\'e VI tau functions. They are obtained from reducing the Hirota bilinear equations that…

The Lie point symmetries of ordinary differential equations (ODEs) that are candidates for having the Painlev\'e property are explored for ODEs of order $n =2, \dots ,5$. Among the 6 ODEs identifying the Painlev\'e transcendents only…

For every system $\{ p_n(z) \}_{n=0}^\infty$ of OPRL or OPUC, we construct Sobolev orthogonal polynomials $y_n(z)$, with explicit integral representations involving $p_n$. Two concrete families of Sobolev orthogonal polynomials (depending…

Consider the Laguerre polynomials and deform them by the introduction in the measure of an exponential singularity at zero. In [Chen Y., Its A., J. Approx. Theory 162 (2010), 270-297, arXiv:0808.3590] the authors proved that this…

We give a survey of the connection between orthogonal polynomials, Toda lattices and related lattices, and Painlev\'e equations (discrete and continuous).

We construct the elliptic Painlev\'e equation and its higher dimensional analogs as the action of line bundles on 1-dimensional sheaves on noncommutative surfaces.

Recurrence coefficients of semi-classical orthogonal polynomials (orthogonal polynomials related to a weight function $w$ such that $w'/w$ is a rational function) are shown to be solutions of non linear differential equations with respect…

It is well-known that differential Painlev\'e equations can be written in a Hamiltonian form. However, a coordinate form of such representation is far from unique -- there are many very different Hamiltonians that result in the same…

We construct a generalisation of what we call Bureau-Guillot systems, i.e. systems of first order equations with coefficient functions being Painlev\'e transcendents. The same Painlev\'e equation is related to the system and it appears as…

We study a fully noncommutative generalisation of the commutative fourth Painlev\'e equation that possesses solutions in terms of an infinite Toda system over an associative unital division ring equipped by a derivation.