Painlevé transcendents

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Painlevé transcendents are solutions to certain nonlinear second order ordinary differential equations which enjoy a desirable property, but which are not generally solvable in terms of elementary functions.

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 monodromy preserving deformations of linear differential equations. One of the most useful classes of special functions are the elliptic functions. These have the property that they are defined by second order ordinary differential equations whose singularities have a remarkable property (known as the Painlevé property): the only movable singularities are poles. This basic property is shared by all linear ordinary differential equations but is rare in nonlinear equations. Around the turn of the nineteenth century, a trio of leading French mathematicians, Paul Painlevé, Émile Picard, and B. Gambier, turned their attention to the classification of the singularity structure of all second order differential equations with polynomial coefficients. In a landmark achievement, they found that up to certain transformations, every such equation can be put into one of fifty canonical forms.

The question then arises: which of these fifty equations are 'irreducible' and define new transcendental functions? In other words which equations cannot be completely solved in terms of previously studied special functions? In a series of papers published from 1893--1902, Painlevé found that in fact forty-four of the fifty equations are reducible, leaving just six equations requiring the introduction of new special functions to solve them. These six second order nonlinear differential equations are now known as the Painlevé equations and their solutions are known as the Painlevé transcendents.

Unfortunately the most general form of the largest equation was missed by Painlevé, but was discovered in 1905 by Richard Fuchs (son of L. Fuchs), by considering monodromy preserving deformations of certain linear Fuchsian equations. It was quickly added to Painlevé's list by Gambier. This is notable since the first five Painlevé equations can be viewed as certain degenerations of the largest (sixth) equation.

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

  • I (Painlevé):
    \frac{d^2y}{dt^2} = 6 y^2 + \lambda t
  • II (Painlevé):
    \frac{d^2y}{dt^2} = 2 y^3 + ty + \mu
  • III (Painlevé):
    ty\frac{d^2y}{dt^2} = t \left(\frac{dy}{dt} \right)^2 -y\frac{dy}{dt} + at + by + cy^3 + dty^4
  • IV (Gambier):
    y\frac{d^2y}{dt^2}= \frac{1}{2} \left(\frac{dy}{dt} \right)^2 -\frac{1}{2}a^2+2(t^2-b)y^2+4ty^3+\frac{3}{2}y^4
  • V (Gambier):
    t^2(y-y^2)\frac{d^2y}{dt^2}= \frac{1}{2}t^2(1-3y) \left( \frac{dy}{dt} \right)^2 -t y (1-y) \frac{dy}{dt} +a y^2 (1-y)^3 + b (1-y)^3 + c ty (1-y) + e t^2 y^2(1+y)
  • VI (R. Fuchs):
    \frac{d^2y}{dt^2}= \frac{1}{2}\left(\frac{1}{y}+\frac{1}{y-1}+\frac{1}{y-t}\right)\left( \frac{dy}{dt} \right)^2 -\left(\frac{1}{t}+\frac{1}{t-1}+\frac{1}{y-t}\right)\frac{dy}{dt} + \frac{y(y-1)(y-t)}{t^2(t-1)^2} \left(\alpha+\beta\frac{t}{y^2}+\gamma\frac{t-1}{(y-1)^2}+\delta\frac{t(t-1)}{(y-t)^2}\right)

The only singularities of the first three are (movable) poles. The last three also possess some fixed singularities.

[edit] References

  • Davis, Harold T. (1962). Introduction to Nonlinear Integral and Differential Equations. New York: Dover. ISBN 0-486-60971-5.  See sections 7.3, chapter 8, and the Appendices
  • Ince, Edward L. (1956). Ordinary Differential Equations. Dover. ISBN 0486603490. 
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