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\section*{Painleve transcendent}

\hypertarget{painlev_transcendents}{}\subsection*{{Painlev\'e{} transcendents}}\label{painlev_transcendents}

In the study of ordinary differential equations one of the important things is the behaviour of [[monodromies]] and closely related singularities of solutions. In the linear case, the poles possibly appear just at the poles of coefficients of the solutions. In the nonlinear case the solutions can appear elsewhere and generally propagate with change of initial conditions. Very important is if the singularities do not move or monodromies don't change with change of parameters. An [[ordinary differential equation]] (ODE) satisfies the \textbf{Painlev\'e{} property} if all solutions are single valued around every movable singularity. A class of such ``good'' nonlinear equations has been defined by \textbf{Paul Painlev\'e{}} (\href{http://en.wikipedia.org/wiki/Paul_Painlev%C3%A9}{wikipedia}), who discovered at the end of 19th century a truly remarkable fact that all second order ODEs of the form

\begin{displaymath}
u'' = F (z; u, u'),
\end{displaymath}
where $F$ is rational in $u,u'$ and analytic in $z$ and which satisfy the Painlev\'e{} property have solutions which can be expressed in terms of well known functions like elementary and hypergeometric functions and only 6 new kinds of \emph{transcendental functions} called \emph{Painlev\'e{} I-VI}. Furthermore he obtained a complete classification of such equations (of second order?) in 50 classes (44+6) up to a number of standard transformations. Painlev\'e{} transcendents are now of central importance in the study of [[integrable systems]].

There are also some noncommutative versions which are still purely understood.

\hypertarget{literature_and_links}{}\subsection*{{Literature and links}}\label{literature_and_links}

\begin{itemize}%
\item P. Painlev\'e{}, \emph{Sur les \'e{}quations differentielles du second ordre et d'ordre superieur, dont l'integrale g\'e{}nerale est uniforme}, Acta Math. \textbf{25} (1902), pp. 1--86.


\item Richard Fuchs, Comptes Rendus de l'Acad\'e{}mie des Sciences Paris 1905 \textbf{141}, 555--558


\item wikipedia: \href{http://en.wikipedia.org/wiki/Painlev%C3%A9_transcendents}{Painlev\'e{} transcendents}


\item \emph{One hundred years of PVI, the Fuchs--Painlev\'e{} equation}, J. Phys. A, special issue, Preface, \href{http://www.researchgate.net/profile/Marta_Mazzocco2/publication/243413584_One_hundred_years_of_PVI_the_FuchsPainlev_equation/links/00b495315eb2aa177d000000}{pdf}


\item Martin D. Kruskal, Nalini Joshi, Rod Halburd, \emph{Analytic and asymptotic methods for nonlinear singularity analysis: a review and extensions of tests for the Painlev\'e{} property}, 1996 \href{http://www.ucl.ac.uk/~ucahrha/Publications/Pond-97.pdf}{pdf}


\item Henryk odek, \emph{The monodromy group}, Monografie Matematyczne \textbf{67}, 588 pp. Birkh\"a{}user 2006


\item A. A. Kapaev, \emph{Quasi-linear Stokes phenomenon for the Painlev\'e{} first equation}, J. Phys. A: Math. Gen. \textbf{37}, 11149 (2004) \href{http://dx.doi.org/10.1088/0305-4470/37/46/005}{doi}


\item A. A. Bolibruch, A. R. Its, A. A. Kapaev, \emph{On the Riemann--Hilbert--Birkhoff inverse monodromy problem and the Painlev\'e{} equations},   , 16:1 (2004), 121--162


\item Marco Bertola, \emph{Fredholm determinants and (noncommutative) Painlev\'e{} II equation}, slides, \href{http://www.impan.pl/~fasde/presentations/Bertola.pdf}{pdf}


\item A. Okounkov, E. Rains, \emph{Noncommutative geometry and Painlevé equations}, Algebra \& Number Theory 9(6) 1363–1400 (2015) \href{https://doi.org/10.2140/ant.2015.9.1363}{doi}



\end{itemize}
Multidimensional generalizations of Painlev\'e{} VI are introduced in

\begin{itemize}%
\item G. Aminov, S. Arthamonov, A. Levin, M. Olshanetsky, A. Zotov, \emph{Painlev\'e{} field theory}, \href{http://arxiv.org/abs/1306.3265}{arxiv/1306.3265}

\end{itemize}
[[!redirects Painlev\%C3\%A9 transcendent]] [[!redirects Painlev\%C3\%A9 transcendents]] [[!redirects Painlevé transcendents]] [[!redirects Painlevé transcendent]] [[!redirects Painlevé property]] [[!redirects Painlevé equation]] [[!redirects Painlevé equations]] [[!redirects Painleve equation]]



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