---
tag: AZ17JCMRKB
kind: concept
name: Painlevé transcendents
msc: ["33E17", 34M55]
sources:
  - url: https://en.wikipedia.org/wiki/Painlev%C3%A9_transcendents
same_as:
  wikidata: Q907724
  wikipedia: Painlevé transcendents
---

## Summary

Painlevé transcendents are solutions of the six Painlevé equations $\mathrm{P_I}$ to $\mathrm{P_{VI}}$, nonlinear second order differential equations with the [[Painlevé property|8EV1G2Y6CC]].
Consider equations $y'' = R(t, y, y')$ with $R$ rational in $y$ and $y'$ that have this property.
Up to changes of variables, the six equations are the only ones among them that are not solvable by earlier known functions.
Their general solutions are new transcendental functions.
They play for nonlinear problems the role that classical special functions play for linear ones, for instance in random matrix theory and integrable systems.

## Explanations

### L1 human

Solutions of nonlinear equations usually have moving singularities whose type depends on the initial values, such as branch points.
Painlevé and his school looked for second order equations whose only moving singularities are poles, so that solutions stay single-valued around them.
Besides equations solvable by known functions, they found exactly six new families.
The simplest is $\mathrm{P_I}$: $y'' = 6y^2 + t$.
Its solutions are [[meromorphic functions|0ZZ7ZYDR8W]] on the whole plane that cannot be written in terms of classical functions.