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In mathematics, '''Painlevé transcendents''' are solutions to certain [[nonlinear]] second-order [[ordinary differential equations|ordinary]] [[complex differential equation|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 were discovered by {{harvs|txt||authorlink=Paul Painlevé|first=Paul |last=Painlevé|year=1900|year2=1902}}, who later became the French prime minister.
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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 functions]]. They are defined by second order ordinary differential equations whose [[mathematical singularity|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 equation or the [[Riccati equation]], which can all be solved explicitly in terms of integration and previously known special functions. [[Charles Émile Picard|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 Painlevé 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
 
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:<math>y^{\prime\prime}=R(y^{\prime},y,t)</math>
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(with ''R'' a rational function) can be put into one of fifty ''canonical forms'' (listed in {{harv|Ince|1956}}).
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{{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 in his work, which were fixed by  B. Gambier and R. Fuchs.) 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.
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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 '''P'''<sup>1</sup> under [[isomonodromic deformation|monodromy-preserving deformations]].  It was  added to Painlevé's list by {{harvs|txt|last=Gambier|year=1910}}.
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{{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.
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==List of Painlevé equations==
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{{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:
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:<math>\frac{d^2y}{dt^2} = 6 y^2 + t </math>
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|2= II (Painlevé):
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:<math>\frac{d^2y}{dt^2} = 2 y^3 + ty + \alpha </math>
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|3= III (Painlevé):
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:<math>ty\frac{d^2y}{dt^2} =  
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:<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 ''z''<sub>0</sub> (where ''h'' 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 ''R'' grows roughly like a constant times ''R''<sup>5/2</sup>.


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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, which also
gives the corresponding degenerations  of the Gauss [[hypergeometric function]]


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


==Hamiltonian systems==
== So Buy Ray Ban Australia Online ==
The Painlevé equations can all be represented as [[Hamiltonian system]]s.


Example: If we put
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:<math>\displaystyle q=y,\quad p=y^{\prime}+y^2+t/2</math>
 
then the second Painlevé equation
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:<math>\displaystyle y^{\prime\prime} =2y^3+ty+b-1/2</math>
 
is equivalent to the Hamiltonian system
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:<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>
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for the Hamiltonian
 
:<math>\displaystyle H=p(p-2q^2-t)/2 -bq.</math>
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==Symmetries==
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A [[Bäcklund transformation]] 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===
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The set of solutions of the type I Painlevé equation
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:<math>y^{\prime\prime}=6y^2+t</math>
is acted on by the order 5 symmetry  ''y''→ζ<sup>3</sup>''y'', ''t''→ζ''t''
where ζ 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===
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In the Hamiltonian formalism of the type II Painlevé equation
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:<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)\rightarrow (q+b/p,p,-b)</math>
and
:<math>\displaystyle (q,p,b)\rightarrow (-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 A<sub>1</sub>; see below).
If ''b''=1/2 then the equation has the solution ''y''=0; applying the Bäcklund transformations generates an infinite family of rational functions that are solutions, such as ''y''=1/''t'', ''y''=2(''t''<sup>3</sup>&minus;2)/''t''(''t''<sup>3</sup>&minus;4), ...


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 P<sub>I</sub>, P<sub>II</sub>, P<sub>III</sub>, P<sub>IV</sub>, P<sub>V</sub>, P<sub>VI</sub> are 0, A<sub>1</sub>, A<sub>1</sub>⊕A<sub>1</sub>, A<sub>2</sub>, A<sub>3</sub>, and D<sub>4</sub>,
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==Relation to other areas==
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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}}.


The Painlevé equations are all reductions of the self dual Yang-Mills equations.
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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.
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*{{springer|id=p/p110040|title=Painlevé-type equations|first=M. |last=Ablowitz}}
 
*{{Citation | 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}}
  <li>[http://www.proyectoalba.com.ar/spip.php?article66/ http://www.proyectoalba.com.ar/spip.php?article66/]</li>
*{{citation|first=J.|last= Chazy |authorlink=Jean Chazy |title=Sur les équations différentielles dont l'intégrale générale possede un coupure essentielle mobile|journal= C.R. Acad. Sci. |place=Paris|volume= 150 |year=1910|pages= 456–458}}
 
*{{citation|first=J.|last= Chazy |title=Sur les équations différentielles de troisième ordre et d'ordre supérieur dont l'intégrale générale a ses points critiques fixés|journal= Acta Math.|volume= 33 |year=1911|pages= 317–385|doi=10.1007/BF02393131}}
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*{{dlmf|id=32|first=P. A. |last=Clarkson}}
 
*{{Citation | editor1-last=Conte | editor1-first=Robert | title=The Painlevé property | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=CRM Series in Mathematical Physics | isbn=978-0-387-98888-7 | mr=1713574 | year=1999 | author=Robert Conte ed.}}
  <li>[http://bbs.qmaker.org/forum.php?mod=viewthread&tid=207180 http://bbs.qmaker.org/forum.php?mod=viewthread&tid=207180]</li>
* {{citation | last=Davis|first= Harold T. | title=Introduction to Nonlinear Integral and Differential Equations | location=New York | publisher=Dover | year=1962 | isbn=0-486-60971-5}} ''See sections 7.3, chapter 8, and the Appendices''
 
*{{Citation | author1-link=Athanassios S. Fokas | last1=Fokas | first1=Athanassios S. | last2=Its | first2=Alexander R. | last3=Kapaev | first3=Andrei A. | last4=Novokshenov | first4=Victor Yu. | title=Painlevé transcendents: The Riemann–Hilbert approach | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Mathematical Surveys and Monographs | isbn=978-0-8218-3651-4 | mr=2264522 | year=2006 | volume=128}}
  <li>[http://yijiansong.u092.iiddcc.com/news/html/?74964.html http://yijiansong.u092.iiddcc.com/news/html/?74964.html]</li>
*{{citation|first=B. |last=Gambier |title=Sur les équations différentielles du second ordre et du premier degré dont l'intégrale générale est à points critique fixés|journal= Acta Math.|volume= 33 |year=1910|pages= 1–55|doi=10.1007/BF02393211}}.
 
*{{Citation | last1=Gromak | first1=Valerii I. | last2=Laine | first2=Ilpo | last3=Shimomura | first3=Shun | title=Painlevé differential equations in the complex plane | publisher=Walter de Gruyter & Co. | location=Berlin | series=de Gruyter Studies in Mathematics | isbn=978-3-11-017379-6 | mr=1960811 | year=2002 | volume=28}}
  </ul>
* {{citation | last=Ince|first= Edward L. | title=Ordinary Differential Equations | publisher=Dover | year=1956 | isbn=0-486-60349-0 }}
*{{Citation | last1=Iwasaki | first1=Katsunori | last2=Kimura | first2=Hironobu | last3=Shimomura | first3=Shun | last4=Yoshida | first4=Masaaki | title=From Gauss to Painlevé | publisher=Friedr. Vieweg & Sohn | location=Braunschweig | series=Aspects of Mathematics, E16 | isbn=978-3-528-06355-9 | mr=1118604 | year=1991}}
*{{Citation | last1=Nishioka | first1=Keiji | title=A note on the transcendency of Painlevé's first transcendent | mr=931951 | year=1988 | journal=Nagoya Mathematical Journal | issn=0027-7630 | volume=109 | pages=63–67}}
*{{Citation | last1=Noumi | first1=Masatoshi | title=Painlevé equations through symmetry | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Translations of Mathematical Monographs | isbn=978-0-8218-3221-9 | mr=2044201 | year=2004 | volume=223}}
*{{Citation | last1=Noumi | first1=Masatoshi | last2=Yamada | first2=Yasuhiko | title=Symmetries in Painlevé equations  | mr=1816984 | year=2004 | journal=Sugaku Expositions | issn=0898-9583 | volume=17 | issue=2 | pages=203–218}}
*{{citation|first=P.|last= Painlevé |title=Memoire sur les équations différentielles dont l'intégrale générale est uniforme|journal= Bull. Soc. Math. Phys. France |volume=28 |year=1900|pages= 201–261}}
*{{citation|first=P.|last= Painlevé |title= Sur les équations différentielles du second ordre et d'ordre supérieur dont l'intégrale générale est uniforme|journal= Acta Math. |volume=25 |year=1902|pages= 1–85|doi=10.1007/BF02419020}}
*{{springer|id=P/p071080|first=N.Kh.|last= Rozov|title=Painlevé equation}}
*{{Citation|last1=Tracy | first1=Craig | last2=Widom | first2=Harold | title=Painlevé functions in statistical physics|
journal=Publ. RIMS Kyoto Univ.|volume=47|year=2011|pages=361–374|
doi=10.2977/PRIMS/38}}
*{{Citation | last1=Umemura | first1=Hiroshi | title=On the irreducibility of Painlevé differential equations | mr=944888 | year=1989 | journal=Sugaku Expositions | volume=2 | issue=2 | pages=231–252}}
*{{Citation | last1=Umemura | first1=Hiroshi | title=Painlevé equations and classical functions | mr=1365704 | year=1998 | journal=Sugaku Expositions | issn=0898-9583 | volume=11 | issue=1 | pages=77–100}}


==External links==
== from natural sources Ray Ban Australia ==
* 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 [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}}


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