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 were discovered by,,, and.
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 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 equation or the Riccati equation, which can all be solved explicitly in terms of integration and previously known special functions.[1] 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
y\prime\prime=R(y\prime,y,t)
(with
R
P1
tried to extend Painlevé's work to higher-order equations, finding some third-order equations with the Painlevé property.
These six equations, traditionally called Painlevé I–VI, are as follows:
The numbers
\alpha
\beta
\gamma
\delta
y
t
The singularities of solutions of these equations are
infty
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
z0
-2 | ||
(z-z | - | |
0) |
z0 | |
10 |
| ||||
(z-z | ||||
0) |
| ||||||||||
(z-z | ||||||||||
0) |
6+ … | |
(z-z | |
0) |
z0
h
R
R5/2
For type II, the singularities are all (movable) simple poles.
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, p. 380), which also gives the corresponding degenerations of the Gauss hypergeometric function (see, p. 372)
III Bessel | ||||||||
\nearrow | \searrow | |||||||
VI Gauss | → | V Kummer | II Airy | → | I None | |||
\searrow | \nearrow | |||||||
IV Hermite–Weber |
The Painlevé equations can all be represented as Hamiltonian systems.
Example: If we put
\displaystyleq=y, p=y\prime+y2+t/2
\displaystyley\prime\prime=2y3+ty+b-1/2
\displaystyleq\prime=
\partialH | |
\partialp |
=p-q2-t/2
\displaystylep\prime=-
\partialH | |
\partialq |
=2pq+b
\displaystyleH=p(p-2q2-t)/2-bq.
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.
The set of solutions of the type I Painlevé equation
y\prime\prime=6y2+t
y\to\zeta3y
t\to\zetat
\zeta
In the Hamiltonian formalism of the type II Painlevé equation
\displaystyley\prime\prime=2y3+ty+b-1/2
\displaystyleq=y,p=y\prime+y2+t/2
\displaystyle(q,p,b)\to(q+b/p,p,-b)
\displaystyle(q,p,b)\to(-q,-p+2q2+t,1-b).
A1
b=1/2
y=0
y=1/t
y=2(t3-2)/t(t3-4)
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
PI
PII
PIII
PIV
PV
PVI
A1
A1 ⊕ A1
A2
A3
D4
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 equations; see .
The Painlevé equations are all reductions of the self-dual Yang–Mills equations; see .
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 conformal blocks at both
c=1
c=infty
c