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In mathematics, '''Painlevé transcendents''' are solutions to certain [[nonlinear differential equation|nonlinear]] [[second-order differential equation|second-order]] [[ordinary differential equation]]s in the complex plane with the '''Painlevé property''' (the only movable singularities are poles), but which are not generally solvable in terms of [[elementary function]]s. They were discovered by
{{harvs|txt|authorlink=Émile Picard|first=Émile|last=Picard|year=1889}},
{{harvs|txt|authorlink=Paul Painlevé|first=Paul |last=Painlevé|year=1900|year2=1902}},
{{harvs|txt|authorlink=:de:Richard Fuchs (Mathematiker)|first=Richard|last=Fuchs|year=1905}}, and
{{harvs|txt|authorlink=:de:Bertrand Gambier|first=Bertrand|last=Gambier|year=1910}}.

==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 deformation]]s of linear differential equations. One of the most useful classes of special functions are the [[elliptic function]]s. They are defined by second-order ordinary differential equations whose [[singularity (mathematics)|singularities]] have the '''Painlevé property''': the only [[movable singularity|movable singularities]] are [[pole (complex analysis)|poles]]. This property is rare in nonlinear equations. Poincaré and L. Fuchs showed that any first order equation with the  Painlevé property can be transformed into the [[Weierstrass elliptic function|Weierstrass elliptic equation]] or the [[Riccati_equation#Complex_analysis|Riccati equation]], which can all be solved explicitly in terms of integration and previously known special functions.<ref>{{cite book | last=Conte | first=Robert | editor-first1=Robert | editor-last1=Conte |title=The Painlevé Property | publisher=Springer New York | publication-place=New York, NY | date=1999 | isbn=978-0-387-98888-7 | doi=10.1007/978-1-4612-1532-5|page=105}}</ref> [[Émile Picard]] pointed out that for orders greater than 1, movable essential singularities can occur, and found a special case of what was later called Painleve VI equation (see below). (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

:<math>y^{\prime\prime}=R(y^{\prime},y,t)</math>

(with <math>R</math> a rational function) can be put into one of fifty ''canonical forms'' (listed in {{harv|Ince|1956}}). {{harvs|last=Painlevé|year1=1900|year2=1902|txt}} found that forty-four of the fifty equations are reducible in the sense that they can be solved in terms of previously known functions, leaving just six equations requiring the introduction of new special functions to solve them. There were some computational errors, and as a result he missed three of the equations, including the general form of Painleve VI.
The errors were fixed and classification completed by Painlevé's student [[:de: Bertrand Gambier|Bertrand Gambier]]. Independently of Painlevé and Gambier, equation Painleve VI was found by [[:de: Richard Fuchs|Richard Fuchs]] from completely different considerations: he studied [[isomonodromic deformation]]s of linear differential equations with [[regular singularities]]. It was a controversial open problem for many years to show that these six equations really were irreducible for generic values of the parameters (they are sometimes reducible for special parameter values; see below), but this was finally proved by {{harvtxt|Nishioka|1988}} and {{harvs|txt|last=Umemura|first= Hiroshi |year=1989}}.
These six second order nonlinear differential equations are called the Painlevé equations and their solutions are called the Painlevé transcendents.

The most general form of the sixth equation was missed by Painlevé, but was discovered in 1905 by Richard Fuchs (son of [[Lazarus Fuchs]]), as the differential equation satisfied by the singularity of a second order Fuchsian equation with 4 regular singular points on the [[projective line]] <math>\mathbf{P}^1</math> under [[isomonodromic deformation|monodromy-preserving deformations]].  It was  added to Painlevé's list by {{harvs|txt|last=Gambier|year=1910}}.

{{harvs|last=Chazy|year1=1910|year2=1911|txt}} tried to extend Painlevé's work to higher-order equations, finding some third-order equations with the Painlevé property.

==List of Painlevé equations==
{{multiple image | direction = vertical | width = 240 | footer = | image1 = PainleveTypeI.svg | alt1 = Painlevé transcendent of the first type | caption1 = | image2 = PainleveTypeII.svg | alt2 = Painlevé transcendent of the second type | caption2 = | image3 = PainleveTypeIII.svg | alt3 = Painlevé transcendent of the third type | caption3 = |}}

These six equations, traditionally called Painlevé I–VI, are as follows:
{{bulleted list

|1= I (Painlevé):

:<math>\frac{d^2y}{dt^2} = 6 y^2 +  t </math>

|2= II (Painlevé):

:<math>\frac{d^2y}{dt^2} = 2 y^3 + ty + \alpha </math>

|3= III (Painlevé):

:<math>\frac{d^2y}{dt^2} = 
\frac{1}{y}\left(\frac{dy}{dt}\right)^2-\frac{1}{t}\frac{dy}{dt} + \frac{1}{t}(\alpha y^2 + \beta) + \gamma y^3 + \frac{\delta}{y}</math>

|4= IV (Gambier):

:<math>\frac{d^2y}{dt^2}=\frac{1}{2y}\left(\frac{dy}{dt}\right)^2+\tfrac32y^3+4ty^2+2(t^2-\alpha)y+\frac{\beta}{y}</math>

|5= V (Gambier):

:<math>\begin{align}
\frac{d^2y}{dt^2}&=
\left(\frac{1}{2 y }+\frac{1}{ y -1}\right) \left( \frac{dy}{dt} \right)^2
-\frac{1}{t} \frac{dy}{dt}\\
&\quad+\frac{( y -1)^2}{t^2}\left(\alpha y +\frac{\beta}{ y }\right) +\gamma\frac{ y }{t}+\delta\frac{ y ( y +1)}{ y -1}\\
\end{align}</math>

|6= VI (R. Fuchs):

:<math>\begin{align}
\frac{d^2y}{dt^2}&=
\frac{1}{2}\left(\frac{1}{y}+\frac{1}{y-1}+\frac{1}{y-t}\right)\left( \frac{dy}{dt} \right)^2
-\left(\frac{1}{t}+\frac{1}{t-1}+\frac{1}{y-t}\right)\frac{dy}{dt} \\&\quad +
\frac{y(y-1)(y-t)}{t^2(t-1)^2}
\left\{\alpha+\beta\frac{t}{y^2}+\gamma\frac{t-1}{(y-1)^2}+\delta\frac{t(t-1)}{(y-t)^2}\right\}\\ 
\end{align}</math>

}}

The numbers <math>\alpha</math>, <math>\beta</math>, <math>\gamma</math>, <math>\delta</math> are complex constants. By rescaling <math>y</math> and <math>t</math> one can choose two of the parameters for type III, and one of the parameters for type V, so these types really have only 2 and 3 independent parameters.

==Singularities==

The singularities of solutions of these equations are 
*The point <math>\infty</math>, and
*The point 0 for types III, V and VI, and
*The point 1 for type VI, and
*Possibly some movable poles

For type I, the singularities are (movable) double poles of residue 0, and the solutions all have an infinite number of such poles in the complex plane. The functions with a double pole at <math>z_0</math> have the Laurent series expansion
:<math>(z-z_0)^{-2}-\frac{z_0}{10}(z-z_0)^2-\frac{1}{6}(z-z_0)^3+h(z-z_0)^4+\frac{z_0^2}{300}(z-z_0)^6+\cdots</math>
converging in some neighborhood of <math>z_0</math> (where <math>h</math> is some complex number). The location of the poles was described in detail by {{harvs|last=Boutroux|year1=1913|year2=1914}}. The number of poles in a ball of radius <math>R</math> grows roughly like a constant times <math>R^{5/2}</math>.

For type II, the singularities are all (movable) simple poles.

==Degenerations==
The first five Painlevé equations are degenerations of the sixth equation. More precisely, some of the equations are degenerations of others according to the following diagram (see {{harvtxt|Clarkson|2006}}, p.&nbsp;380), which also gives the corresponding degenerations of the Gauss [[hypergeometric function]] (see {{harvtxt|Clarkson|2006}}, p.&nbsp;372)

{|
|
|
|
|
|&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;III [[Bessel function|Bessel]]
|-
|
|
|
|<math>\nearrow</math>
|
|<math>\searrow</math>
|-
|VI [[Hypergeometric function|Gauss]]
|→
|V [[Confluent hypergeometric function|Kummer]]
|
|
|
|II [[Airy function|Airy]]
|→
|I None
|-
|
|
|
|<math>\searrow</math>
|
|<math>\nearrow</math>
|-
|
|
|
|
|IV [[Hermite–Weber function|Hermite–Weber]]
|}

==Hamiltonian systems==
The Painlevé equations can all be represented as [[Hamiltonian system]]s.

Example: If we put
:<math>\displaystyle q=y,\quad p=y^{\prime}+y^2+t/2</math>
then the second Painlevé equation 
:<math>\displaystyle y^{\prime\prime} =2y^3+ty+b-1/2</math>
is equivalent to the Hamiltonian system
:<math>\displaystyle q^{\prime}=\frac{\partial H}{\partial p} = p-q^2-t/2</math>
:<math>\displaystyle p^{\prime}=-\frac{\partial H}{\partial q} = 2pq+b</math>
for the Hamiltonian 
:<math>\displaystyle H=p(p-2q^2-t)/2 -bq.</math>

==Symmetries==
A [[Bäcklund transform]] is a transformation of the dependent and independent variables of a differential equation that transforms it to a similar equation. The Painlevé equations all have discrete groups of Bäcklund transformations acting on them, which can be used to generate new solutions from known ones.

===Example type I===

The set of solutions of the type I Painlevé equation 
:<math>y^{\prime\prime}=6y^2+t</math>
is acted on by the order 5 symmetry <math>y\to\zeta^3 y</math>, <math>t\to\zeta t</math> where <math>\zeta</math> is a fifth root of 1. There are two solutions invariant under this transformation, one with a pole of order 2 at 0, and the other with a zero of order 3 at 0.

===Example type II===

In the Hamiltonian formalism of the type II Painlevé equation
:<math>\displaystyle y^{\prime\prime}=2y^3+ty+b-1/2</math>
with 
:<math>\displaystyle q=y,p=y^\prime+y^2+t/2</math>
two Bäcklund transformations are given by
:<math>\displaystyle (q,p,b)\to (q+b/p,p,-b)</math>
and
:<math>\displaystyle (q,p,b)\to (-q, -p+2q^2+t,1-b).</math>
These both have order 2, and generate an [[infinite dihedral group]] of Bäcklund transformations (which is in fact the affine Weyl group of <math>A_1</math>; 
see below).
If <math>b=1/2</math> then the equation has the solution <math>y=0</math>; applying the Bäcklund transformations generates an infinite family of rational functions that are solutions, such as <math>y=1/t</math>, <math>y=2(t^3-2)/t(t^3-4)</math>, ...

Okamoto discovered that the parameter space of each Painlevé equation can be identified with the [[Cartan subalgebra]] of a [[semisimple Lie algebra]], such that actions of the [[affine Weyl group]] lift to Bäcklund transformations of the equations. The Lie algebras for 
<math>P_I</math>, <math>P_{II}</math>, <math>P_{III}</math>, <math>P_{IV}</math>, <math>P_V</math>, <math>P_{VI}</math> 
are 0, <math>A_1</math>, <math>A_1\oplus A_1</math>, <math>A_2</math>, <math>A_3</math>, and <math>D_4</math>.

==Relation to other areas==

One of the main reasons Painlevé equations are studied is their relation with invariance of the [[monodromy]] of linear systems with [[regular singularities]] under changes in the locus of the poles. In particular, Painlevé VI was discovered by Richard Fuchs because of this relation. This subject is described in the article on [[isomonodromic deformation]].

The Painlevé equations are all reductions of integrable [[partial differential equation]]s; see {{harvs| last1=Ablowitz | first1=M. J. | last2=Clarkson | first2=P. A. | title=Solitons, nonlinear evolution equations and inverse scattering | publisher=Cambridge University Press | series=London Mathematical Society Lecture Note Series | isbn=978-0-521-38730-9 | mr=1149378 | year=1991 | volume=149 | txt}}.

The Painlevé equations are all reductions of the [[self-dual Yang–Mills equations]]; see {{harvs|last1=Ablowitz|last2=Chakravarty|last3=Halburd|year=2003|txt}}.

The Painlevé transcendents appear in [[random matrix theory]] in the formula for the [[Tracy–Widom distribution]], the 2D [[Ising model]], the [[asymmetric simple exclusion process]] and in two-dimensional quantum gravity.

The Painlevé VI equation appears in [[two-dimensional conformal field theory]]: it is obeyed by combinations of [[Virasoro conformal block|conformal blocks]] at both <math>c=1</math> and <math>c=\infty</math>, where <math>c</math> is the central charge of the [[Virasoro algebra]].

==Notes==
{{reflist}}
==References==
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==External links==
* Clarkson, P. A. [http://dlmf.nist.gov/32 Painlevé Transcendents], Chapter 32 of the NIST [http://dlmf.nist.gov/ Digital Library of Mathematical Functions]
* Joshi, Nalini [http://www.maths.usyd.edu.au/u/nalini/p0.html What is this thing called Painlevé?]
* Takasaki, Kanehisa [https://web.archive.org/web/20080406003354/http://www.math.h.kyoto-u.ac.jp/~takasaki/soliton-lab/chron/painleve.html Painlevé Equations]
* {{MathWorld|urlname=PainleveTranscendents|title=Painleve Transcendents}}
* {{MathWorld|urlname=PainleveProperty|title=Painleve Property}}

{{DEFAULTSORT:Painleve Transcendents}}
[[Category:Special functions]]
[[Category:Ordinary differential equations]]