Painlevé transcendents

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

History

Origins

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 simple poles. This property is rare in nonlinear equations.

Poincaré and Lazarus Fuchs showed that any first order equation (that is, an ODE involving only up to the first derivative) with the Painlevé property can be transformed into the Weierstrass elliptic equation or the Riccati equation, all of which can 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 found in 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.) Specifically, let <math display="inline">\varphi</math> be the elliptic function defined by<math display="block">

\varphi: y \mapsto \varphi(y, x), \qquad y=\int_{\infty}^{\varphi} \frac{\mathrm{d} z}{\sqrt{z(z-1)(z-x)</math>

and let <math display="inline">\omega_1(x), \omega_2(x)</math> be its two half-periods. Then the function <math display="block">

u: x \mapsto u(x)=\varphi\left(2 c_1 \omega_1(x)+2 c_2 \omega_2(x), x\right)

</math>with <math display="inline">\left(c_1, c_2\right)</math> arbitrary constants satisfies the Painleve VI equation in the case of <math display="inline">\alpha=\beta=\gamma= \delta - 1/2 = 0</math>.

Classification

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

:<math>y^{\prime\prime}=R(y^{\prime},y,t)</math>

(with <math>R</math> a rational function) can be put into one of 50 canonical forms (listed in ).

found that 44 of the 50 equations are reducible, in the sense that they can be solved in terms of previously known functions, leaving just 6 equations requiring the introduction of new special functions to solve them. These six second order nonlinear differential equations are called the Painlevé equations and their solutions are called the Painlevé transcendents. There were some computational errors, and as a result he missed 3 of the equations, including the general form of Painleve VI. Painlevé's student Bertrand Gambier fixed the errors and completed the classification.

Independently of Painlevé and Gambier, equation Painleve VI was found by Richard Fuchs from completely different considerations: he studied isomonodromic deformations of linear differential equations with regular singularities.

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 the projective line <math>\mathbf{P}^1</math> under monodromy-preserving deformations. It was added to Painlevé's list by .

Subsequent work

tried to extend Painlevé's work to higher-order equations, finding some third-order equations with the Painlevé property.

It was an open problem for many years to show that these 6 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 .

List of Painlevé equations

These six equations, traditionally called Painlevé I–VI, are as follows:

The symbols <math>\alpha</math>, <math>\beta</math>, <math>\gamma</math>, <math>\delta</math> denote complex-valued constants.

If <math display="inline">\gamma \delta \neq 0</math> in <math display="inline">\mathrm{P}_{\mathrm{III</math>, then one can set <math display="inline">\gamma=1</math> and <math display="inline">\delta=-1</math>, without loss of generality, by rescaling <math display="inline">\alpha</math> and <math display="inline">\beta</math> if necessary. If <math display="inline">\gamma=0</math> and <math display="inline">\alpha \delta \neq 0</math> in <math display="inline">\mathrm{P}_{\mathrm{III</math>, then set <math display="inline">\alpha=1</math> and <math display="inline">\delta=-1</math>, without loss of generality. Lastly, if <math display="inline">\delta=0</math> and <math display="inline">\beta \gamma \neq 0</math>, then one can set <math display="inline">\beta=-1</math> and <math display="inline">\gamma=1</math>, without loss of generality.

If <math display="inline">\delta \neq 0</math> in <math display="inline">\mathrm{P}_{\mathrm{V</math>, then one can set <math display="inline">\delta=-\frac{1}{2}</math>, without loss of generality.

Singularities

The singularities of solutions of these equations are

The point <math>\infty</math>, and
The point 0 for types III, V and VI, and
The point 1 for type VI, and
Possibly some movable poles

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 <math>z_0</math> have the Laurent series expansion

:<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 <math>z_0</math> (where <math>h</math> is some complex number). The location of the poles was described in detail by . The number of poles in a ball of radius <math>R</math> grows roughly like a constant times <math>R^{5/2}</math>.

For type II, the singularities are all (movable) simple poles.

Asymptotics

I

thumb|Painlevé I solutions for <math display="inline">y(0)=0</math> and <math display="inline">y(0)=0</math>, for various values of <math>k</math>, with the asymptotic parabola.

There are solutions of Painlevé I such that<math display="block">

y(t)=-\sqrt{\tfrac{1}{6}|t|}+d|t|^{-1/8}\sin\left(\phi(t)-\theta_{0}\right)+o\left(|t|^{-1/8}\right)

</math>where<math display="block">

\phi(t)=(24)^{1/4}\left(\tfrac{4}{5}|t|^{5/4}-\tfrac{5}{8}d^{2}\ln|t|\right)

</math>and <math display="inline">d</math> and <math display="inline">\theta_{0}</math> are constants. There are also solutions such that <math display="block">

y(t)\sim\sqrt{\tfrac{1}{6}|t|}

</math>although such solutions are unstable under perturbation.

For given initial conditions <math display="inline">y(0)=0</math> and <math display="inline">y^{\prime}(0)=k</math>, with <math display="inline">k</math> real, <math display="inline">y(t)</math> has at least one pole on the real axis. There are two special values of <math display="inline">k</math>, <math display="inline">k_{1}</math> and <math display="inline">k_{2}</math>, with the properties <math display="inline">-0.451428<k_{1}<-0.451427</math>, <math display="inline">1.851853<k_{2}<1.851855</math>, such that if <math>k \in (k_1, k_2)</math> then the solution oscillates about, and is asymptotic to, <math>-\sqrt{|t|/6}</math>.

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 (see , p. 380), which also gives the corresponding degenerations of the Gauss hypergeometric function (see , p. 372)

|      III Bessel

|<math>\nearrow</math>

|<math>\searrow</math>

|VI Gauss

|→

|V Kummer

|II Airy

|→

|I None

|<math>\searrow</math>

|<math>\nearrow</math>

|IV Hermite–Weber

Hamiltonian systems

The Painlevé equations can all be represented as Hamiltonian systems.

Example: If we put

:<math>\displaystyle q=y,\quad p=y^{\prime}+y^2+t/2</math>

then the second Painlevé equation

:<math>\displaystyle y^{\prime\prime} =2y^3+ty+b-1/2</math>

is equivalent to the Hamiltonian system

:<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>

for the Hamiltonian

:<math>\displaystyle H=p(p-2q^2-t)/2 -bq.</math>

Symmetries

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.

Example type I

The set of solutions of the type I Painlevé equation

:<math>y^{\prime\prime}=6y^2+t</math>

is acted on by the order 5 symmetry <math>y\to\zeta^3 y</math>, <math>t\to\zeta t</math> where <math>\zeta</math> 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 equation

:<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)\to (q+b/p,p,-b)</math>

and

:<math>\displaystyle (q,p,b)\to (-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 <math>A_1</math>;

see below).

If <math>b=1/2</math> then the equation has the solution <math>y=0</math>; applying the Bäcklund transformations generates an infinite family of rational functions that are solutions, such as <math>y=1/t</math>, <math>y=2(t^3-2)/t(t^3-4)</math>, ...

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

are 0, <math>A_1</math>, <math>A_1\oplus A_1</math>, <math>A_2</math>, <math>A_3</math>, and <math>D_4</math>.

Special parameter values

<math>P_{\mathrm{I</math> has no free parameters. For the other cases, they are irreducible for generic parameter values, but for special parameter values they admit solutions that satisfy a first-order Riccati equation, which can be expressed in terms of special functions:

rational solutions for 2, 3, 4, 5, 6;
algebraic solutions for 3, 5, 6;
other special-function solutions: Airy for 2, Bessel for 3, parabolic-cylinder for 4, Whittaker for 5, and hypergeometric for 6.

Relation to other areas

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.

<math>P_{\mathrm{II</math> has been used to describe a dynamic passage through a quantum phase transition, leading to precise scaling of excitations and revealing relations to integrable multistate Landau-Zener models. Integrable generalizations of <math>P_{\mathrm{II</math> have been found based on this application.

<math>P_{\mathrm{VI</math> appears in two-dimensional conformal field theory: it is obeyed by combinations of conformal blocks at both <math>c=1</math> and <math>c=\infty</math>, where <math>c</math> is the central charge of the Virasoro algebra.

Notes

References

Robert M. M. Conte:The Painlevé Handbook, Springer, ISBN 978-9400796270, (2014).
Robert M. M. Conte:The Painlevé Handbook, Springer; 2nd ed, ISBN 978-3030533397, (2022).
See sections 7.3, chapter 8, and the Appendices
.
Martin A. Guest, Claus Hertling: Painlevé III: A Case Study in the Geometry of Meromorphic Connections, Springer, LNM, vol.2198, ISBN 9783319665269, (2017).
Alexander R. Its, Victor Yu. Novokshenov:The Isomonodromic Deformation Method in the Theory of Painlevé Equations, Springer, LNM 1191, ISBN 9783540398233, (1986).

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