Painlevé transcendents

In mathematics, Painlevé transcendents are solutions to certain nonlinear second-order ordinary differential equations in the complex plane with the Painlevé property (the only movable singularities are poles), but which are not generally solvable in terms of elementary functions. They are named for the French mathematican (and prime minister) Paul Painlevé who found them around 1900.

History

Painlevé transcendents have their origin in the study of special functions, which often arise as solutions of differential equations, as well as in the study of isomonodromic deformations of linear differential equations. One of the most useful classes of special functions are the elliptic functions. They are defined by second order ordinary differential equations whose singularities have the Painlevé property: the only movable singularities are poles. This property is shared by all linear ordinary differential equations but is rare in nonlinear equations. Poincare and L. Fuchs showed that any first order equation with the Painlevé property can be transformed into the Weierstrass equation or the Riccati equation, which can all be solved explicitly in terms of integration and previously known special functions. Émile Picard pointed out that for orders greater than 1, movable essential singularities can occur, and tried and failed to find new examples with the Painleve property. (For orders greater than 2 the solutions can have moving natural boundaries.) Around 1900, Paul Painlevé studied second order differential equations with no movable singularities. He found that up to certain transformations, every such equation of the form

: $y^{primeprime}=R(y^{prime},y,t)$

(with "R" a rational function) can be put into one of fifty "canonical forms" (listed in | year=1991 | volume=149.

The Painlevé equations are all reductions of the self dual Yang-Mills equations.

References

External links

*dlmf|id=32
*Kanehisa Takasaki [http://www.math.h.kyoto-u.ac.jp/~takasaki/soliton-lab/chron/painleve.html Painlevé Equations]
*
*

Wikimedia Foundation. 2010.

Игры ⚽ Поможем написать реферат

Look at other dictionaries:

Paul Painlevé — Mandats 95e 115e président du Conseil des ministres français … Wikipédia en Français
Paul Painlevé — Infobox Prime Minister name=Paul Painlevé order=84th Prime Minister of France term start =12 September 1917 term end =16 November 1917 president =Raymond Poincaré predecessor =Alexandre Ribot successor =Georges Clemenceau order2=92nd Prime… … Wikipedia
Isomonodromic deformation — In mathematics, the equations governing the isomonodromic deformation of meromorphic linear systems of ordinary differential equations are, in a fairly precise sense, the most fundamental exact nonlinear differential equations. As a result, their … Wikipedia
Charles Émile Picard — Born 24 July 1856(1856 07 24) Paris … Wikipedia
List of special functions and eponyms — This is a list of special function eponyms in mathematics, to cover the theory of special functions, the differential equations they satisfy, named differential operators of the theory (but not intended to include every mathematical eponym).… … Wikipedia
Complex differential equation — A complex differential equation is a differential equation whose solutions are functions of a complex variable. Constructing integrals involves choice of what path to take, which means singularities and branch points of the equation need to be… … Wikipedia
List of mathematical functions — In mathematics, several functions or groups of functions are important enough to deserve their own names. This is a listing of pointers to those articles which explain these functions in more detail. There is a large theory of special functions… … Wikipedia
List of mathematics articles (P) — NOTOC P P = NP problem P adic analysis P adic number P adic order P compact group P group P² irreducible P Laplacian P matrix P rep P value P vector P y method Pacific Journal of Mathematics Package merge algorithm Packed storage matrix Packing… … Wikipedia
Athanassios S. Fokas — (b. June 30, 1952) is a Greek mathematician, well known in the field of integrable nonlinear partial differential equations. Fokas was born on the Greek island of Kefallonia. Academic qualifications* BSc (Aeronautics) Imperial College (1975) *… … Wikipedia

Academic Dictionaries and Encyclopedias

Painlevé transcendents

Look at other dictionaries:

Share the article and excerpts

Academic Dictionaries and Encyclopedias

Wikipedia

Painlevé transcendents

Look at other dictionaries:

Share the article and excerpts

Direct link