Painlevé Classification of a Class of Differential Equations of the Second Order and Second Degree

In this paper we construct all Painlevé-type differential equations of the form (d²y/dx²)² = F(x,y,dy/dx), where F is rational in y and y′=dy/dx, locally analytic in x, and not a perfect square. No further simplifying assumptions are made, but it is found that the absence of a term linear in y″ in the class of equations under investigation forces F to be a polynomial in y and y′. We find exactly six distinct classes of second-degree Painlevé equations, denoted SD-I,??,SD-VI, some of which further subdivide into canonical subcases. Only the first three classes (or at least equations transformable to the first three classes) and part of the sixth have appeared previously in the literature, especially the work of Chazy and Bureau. The fourth and fifth classes are new. The unified treatment of SD-I, which we call the “master Painlevé equation,” is new. Complete solutions are given in terms of the classical Painlevé transcendents, elliptic functions, or solutions of linear equations. In an appendix, it is shown that a class of second-degree equations generalizing the Appell equation can always be reduced to a second-order linear equation.     

Painlevé Classification of All Semilinear Partial Differential Equations of the Second Order. II. Parabolic and Higher Dimensional Equations

This paper extends the work of the previous paper (I) on the Painlevé classification of second-order semilinear partial differential equations to the case of parabolic equations in two independent variables, u_xx = F(x, y, u, u_x, u_y), and irreducible equations in three or more independent variables of the form, Σ_ijR_ij (x₁,…, x_n)u,_ij = F(x₁,…, x_n; u,₁,…, u,_n). In each case, F is assumed to be rational in u and its first derivatives and no other simplifying assumptions are made. In addition to the 22 hyperbolic equations found in paper I, we find 10 equivalence classes of parabolic equations with the Painlevé property, denoted PS-I, PS-I1,…, PS-X, equation PS-II being a generalization of Burgers' equation denoted the Forsyth-Burgers equation, and 13 higher-dimensional Painlevé equations, denoted GS-I, GS-II,…, GS-XIII. The lists are complete up to the equivalence relation of Möbius transformations in u and arbitrary changes of the independent variables. In order to avoid repetition, the proofs are sketched very briefly in cases where they closely resemble those for the corresponding hyperbolic problem. Every equation is solved by transforming to a linear partial differential equation, from which it follows that there are no non trivial soliton equations among the two classes of Painlevé equations treated in this paper.     

Distributions of Poles to Painlevé Transcendents via Padé Approximations

A version of the Fair–Luke algorithm has been used to find the Padé approximate solutions to the Painlevé I, II, and IV equations. The distributions of poles in the complex plane are studied to check the dynamics of movable poles and the emergence of rational and truncated solutions, as well as various patterns formed by the poles. The high-order approximations allow us to check asymptotic expansions at infinity and estimate the range of asymptotic domains. The Coulomb gas interpretation of the pole ensembles is discussed in view of the patterns arising in Painlevé IV transcendents.     

Rational Solutions of Painlevé Equations

With Bäcklund transformations, we construct explicit solutions of Painlevé equations 2 and 4. Independently, we find solutions of degenerate cases of equations 3 and 5. The six Painlevé transcendents are referred to as 1–6.     

Asymptotics of Discrete Painlevé V transcendents via the Riemann–Hilbert Approach

We study a system of discrete Painlevé V equations via the Riemann–Hilbert approach. We begin with an isomonodromy problem for dPV, which admits a discrete Riemann–Hilbert problem formulation. The asymptotics of the discrete Riemann–Hilbert problem is derived via the nonlinear steepest descent method of Deift and Zhou. In the analysis, a parametrix is constructed in terms of specific Painlevé V transcendents. As a result, the asymptotics of the dPV transcendents are represented in terms of the PV transcendents. In the special case, our result confirms a conjecture of Borodin, that the difference Schlesinger equations converge to the differential Schlesinger equations at the solution level.     

Painlevé IV with both Parameters Zero: A Numerical Study

The six Painlevé equations were introduced over a century ago, motivated by rather theoretical considerations. Over the last several decades, these equations and their solutions, known as the Painlevé transcendents, have been found to play an increasingly central role in numerous areas of mathematical physics. Due to extensive dense pole fields in the complex plane, their numerical evaluation remained challenging until the recent introduction of a fast “pole field solver” [ 1 ]. The fourth Painlevé equation has two free parameters in its coefficients, as well as two free initial conditions. The present study applies this new computational tool to the special case when both of its parameters are zero. We confirm existing analytic and asymptotic knowledge about the equation, and also explore solution regimes which have not been described in the previous literature.     

The Yosida class is universal

We discuss families of meromorphic functions f _h obtained from single functions f by the re-scaling process f _h (z) = h ^−α f (h + h ^−β z), generalising Yosida’s process f _h (z) = f (h + z). The main objective is to obtain information about the value distribution of the generating functions f themselves. Among the most prominent (generalised) Yosida functions are the elliptic functions and also some first, second and fourth Painlevé transcendents. The Yosida class A ₀ contains all limit functions of generalised Yosida functions-the Yosida class is universal.     

Uniqueness theorems for solutions of Painlevé transcendents

The paper deals with the uniqueness problems when two meromorphic functions f and g share three distinct values CM and f satisfies the first, second or fourth Painlevé transcendents.     

Paul Painlevé and his contribution to science

The life and career of the great French mathematician and politician Paul Painlevé is described. His contribution to the analytical theory of nonlinear differential equations was significant. The paper outlines the achievements of Paul Painlevé and his students in the investigation of an interesting class of nonlinear second-order equations and new equations defining a completely new class of special functions, now called the Painlevé transcendents. The contribution of Paul Painlevé to the study of algebraic nonintegrability of the N-body problem, his remarkable observations in mechanics, in particular, paradoxes arising in the dynamics of systems with friction, his attempt to create the axiomatics of mechanics and his contribution to gravitation theory are discussed.     

A Direct Proof that Solutions of the Six Painlevé Equations Have No Movable Singularities Except Poles

The Painlevé property of an nth-order differential equation is that no solution has any movable singularities other than poles. This property is strongly indicative of complete integrability (the existence of n ? 1 integrals). However, the usual technique employed to test for the Painlevé property seeks only movable algebraic (or logarithmic) singularities. More general singularities are ignored. But, the six standard Painlevé equations are known to have no such singularities. Painlevé's proof of this is long and laborious; we give here a direct proof.     

Essential singularity of the second-genus Painlevé function and the nonlinear stokes phenomenon

By the isomonodromic deformation method, the leading term of the elliptic asymptotics as x→∞ of the solution of the second Painlevé equation is constructed in the generic case. The equations for the modulus of this elliptic sine (which depends only on arg x) are given. The phase of the elliptic sine for any arg x is explicitly expressed in terms of first integrals of the Painlevé equation, i.e., in terms of the Stokes multipliers of the associated linear system. A nonlinear Stokes phenomenon typical for the asymptotic behavior of the Painlevé function is described. Bibliography: 25 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 187, pp. 139–170, 1990. Translated by O. A. Ivanov.     

Modifications of bundles, elliptic integrable systems, and related problems

We describe a construction of elliptic integrable systems based on bundles with nontrivial characteristic classes, especially attending to the bundle-modification procedure, which relates models corresponding to different characteristic classes. We discuss applications and related problems such as the Knizhnik-Zamolodchikov-Bernard equations, classical and quantum R-matrices, monopoles, spectral duality, Painlevé equations, and the classical-quantum correspondence. For an SL(N,?)-bundle on an elliptic curve with nontrivial characteristic classes, we obtain equations of isomonodromy deformations.     

A general framework for solving Riemann–Hilbert problems numerically

A new, numerical framework for the approximation of solutions to matrix-valued Riemann?CHilbert problems is developed, based on a recent method for the homogeneous Painlevé II Riemann?CHilbert problem. We demonstrate its effectiveness by computing solutions to other Painlevé transcendents. An implementation in Mathematica is made available online.     

Problèmes de modules pour les formes différentielles singulières dans le plan complexe

Resumé On considère des équations de Pfaff holomorphes à l’origine de ℂ², ω=a(x, y)dx +b(x, y)dy. Sous des hypothèses génériques, portant sur le premier jet non nulω _v deω, on décrit explicitement l’espace des modules de ω pourv petit. On s’intéresse aussi aux formes rigides et aux problèmes sous-jacents à ce type de question, notemment l’invariance topologique de l’holonomie projective.

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We consider holomorphic Pfaffian equations ω=a(x, y)dx +b(x, y)dy. Under generic assumptions on the first significant jet of ω, we describe the space of moduli for Pfaffian equations of small order. Problems of rigidity and topological invariance of projective holonomy are also studied.

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Une partie de ce travail a été réalisé lors d’un séjour du premier auteur à l’IMPA de Rio et à l’Université Fédérale du Minas Gerais. Ceci grace au concours du CNPQ (Brésil) et du Ministère des Relations Extérieures (France).     

Value distribution of Painlevé transcendents of the first and the second kind

We examine value distribution properties of the first and the second Painlevé transcendents. For every transcendental meromorphic solution ϕ(z) (resp. ψ(z)) of the first (resp. second) Painlevé equation, the deficiency δ(g,ϕ) (resp. δ(g, ψ)) of a small functiong(z) does not exceed 1/2. Furthermore, for ϕ(z), the ramification index satisfies ϑ(gφ)≤5/12.