Painlevé transcendents
Encyclopedia
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 , who later became the French prime minister.
, 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 singularities
have the Painlevé property: the only movable singularities
are poles. This property is shared by all linear ordinary differential equations but 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. 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 Painleve 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
of the form
(with R a rational function) can be put into one of fifty canonical forms (listed in ).
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 and .
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 P1 under monodromy-preserving deformations
. It was added to Painlevé's list by .
tried to extend Painlevé's work to higher order equations, finding some third order equations with the Painlevé property.
The numbers α, β, γ, δ are complex constants. By rescaling y and t 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.
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 have the Laurent series expansion
converging in some neighborhood of z0 (where h is some complex number). The location of the poles was described in detail by . The number of poles in a ball of radius R grows roughly like a constant times R5/2.
For type II, the singularities are all (movable) simple poles.
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
s.
Example: If we put
then the second Painlevé equation
is equivalent to the Hamiltonian system
for the Hamiltonian
Bäcklund transformations acting on them, which can be used to generate new solutions from known ones.
is acted on by the order 5 symmetry y→ζ3y, t→ζt
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.
with
two Bäcklund transformations are given by
and
These both have order 2, and generate an infinite dihedral group
of Bäcklund transformations (which is in fact the affine Weyl group of A1; 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(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 are 0, A1, A1⊕A1, A2, A3, and D4,
s; see .
The Painlevé equations are all reductions of the self dual Yang-Mills equations.
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.
Complex differential equation
A complex differential equation is a differential equation whose solutions are functions of a complex variable.Constructing integrals involves choice of what path to take, which means singularities and branch points of the equation need to be studied...
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 , who later became the French prime minister.
History
Painlevé transcendents have their origin in the study of special functionsSpecial functions
Special functions are particular mathematical functions which have more or less established names and notations due to their importance in mathematical analysis, functional analysis, physics, or other applications....
, which often arise as solutions of differential equations, as well as in the study of isomonodromic deformation
Isomonodromic deformation
In mathematics, the equations governing the isomonodromic deformation of meromorphic linear systems of ordinary differential equations are, in a fairly precise sense, the most fundamental exact nonlinear differential equations...
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 singularities
Mathematical singularity
In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability...
have the Painlevé property: the only movable singularities
Movable singularity
In the theory of ordinary differential equations, a movable singularity is a point where the solution of the equation behaves badly and which is "movable" in the sense that its location depends on the initial conditions of the differential equation....
are poles. This property is shared by all linear ordinary differential equations but 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
Riccati equation
In mathematics, a Riccati equation is any ordinary differential equation that is quadratic in the unknown function. In other words, it is an equation of the form y' = q_0 + q_1 \, y + q_2 \, y^2...
, which can all be solved explicitly in terms of integration and previously known special functions. Émile Picard
Charles Émile Picard
Charles Émile Picard FRS was a French mathematician. He was elected the fifteenth member to occupy seat 1 of the Académie Française in 1924.- Biography :...
pointed out that for orders greater than 1, movable essential singularities can occur, and tried and failed to find new examples with the Painleve property. (For orders greater than 2 the solutions can have moving natural boundaries.) Around 1900, Paul Painlevé
Paul Painlevé
Paul Painlevé was a French mathematician and politician. He served twice as Prime Minister of the Third Republic: 12 September – 13 November 1917 and 17 April – 22 November 1925.-Early life:Painlevé was born in Paris....
studied second order differential equations with no movable singularities. He found that up to certain transformations, every such equation
of the form
(with R a rational function) can be put into one of fifty canonical forms (listed in ).
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 and .
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
Lazarus Fuchs
Lazarus Immanuel Fuchs was a German mathematician who contributed important research in the field of linear differential equations...
), as the differential equation satisfied by the singularity of a second order Fuchsian equation with 4 regular singular points on P1 under monodromy-preserving deformations
Isomonodromic deformation
In mathematics, the equations governing the isomonodromic deformation of meromorphic linear systems of ordinary differential equations are, in a fairly precise sense, the most fundamental exact nonlinear differential equations...
. It was added to Painlevé's list by .
tried to extend Painlevé's work to higher order equations, finding some third order equations with the Painlevé property.
List of Painlevé equations
These six equations, traditionally called Painlevé I-VI, are as follows:- I (Painlevé):
- II (Painlevé):
- III (Painlevé):
- IV (Gambier):
- V (Gambier):
- VI (R. Fuchs):
The numbers α, β, γ, δ are complex constants. By rescaling y and t 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 possible singularities of these equations are- Movable poles
- The point ∞
- The point 0 for types III, V and VI
- The point 1 for type VI
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 have the Laurent series expansion
converging in some neighborhood of z0 (where h is some complex number). The location of the poles was described in detail by . The number of poles in a ball of radius R grows roughly like a constant times R5/2.
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, which also
gives the corresponding degenerations of the Gauss hypergeometric function
| III Bessel Bessel function In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y of Bessel's differential equation:... |
||||||||
 |
 |
|||||||
| VI Gauss | → | V Kummer Confluent hypergeometric function In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three regular singularities merge into an irregular singularity... |
II Airy Airy function In the physical sciences, the Airy function Ai is a special function named after the British astronomer George Biddell Airy... |
→ | I None | |||
 |
 |
|||||||
| IV Hermite-Weber |
Hamiltonian systems
The Painlevé equations can all be represented as Hamiltonian systemHamiltonian system
In physics and classical mechanics, a Hamiltonian system is a physical system in which forces are momentum invariant. Hamiltonian systems are studied in Hamiltonian mechanics....
s.
Example: If we put
then the second Painlevé equation
is equivalent to the Hamiltonian system
for the Hamiltonian
Symmetries
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 ofBäcklund 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 equationis acted on by the order 5 symmetry y→ζ3y, t→ζt
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
In the Hamiltonian formalism of the type II Painlevé equationwith
two Bäcklund transformations are given by
and
These both have order 2, and generate an infinite dihedral group
Infinite dihedral group
In mathematics, the infinite dihedral group Dih∞ is an infinite group with properties analogous to those of the finite dihedral groups.-Definition:...
of Bäcklund transformations (which is in fact the affine Weyl group of A1; 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(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 are 0, A1, A1⊕A1, A2, A3, and D4,
Relation to other areas
The Painlevé equations are all reductions of integrable partial differential equationPartial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
s; see .
The Painlevé equations are all reductions of the self dual Yang-Mills equations.
The Painlevé transcendents appear in random matrix theory in the formula for the Tracy–Widom distribution
Tracy–Widom distribution
The Tracy–Widom distribution, introduced by , is the probability distribution of the largest eigenvalue of a random hermitian matrix in the edge scaling limit. It also appears in the distribution of the length of the longest increasing subsequence of random permutations and in current fluctuations...
, the 2D Ising model
Ising model
The Ising model is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables called spins that can be in one of two states . The spins are arranged in a graph , and each spin interacts with its nearest neighbors...
, the asymmetric simple exclusion process and in two-dimensional quantum gravity.
External links
- Clarkson, P.A. Painlevé Transcendents, Chapter 32 of the NIST Digital Library of Mathematical Functions
- Joshi, Nalini What is this thing called Painlevé?
- Takasaki, Kanehisa Painlevé Equations