# §32.2 Differential Equations

## §32.2(i) Introduction

with , , , and arbitrary constants. The solutions of  are called the Painlevé transcendents. The six equations are sometimes referred to as the Painlevé transcendents, but in this chapter this term will be used only for their solutions.

Let

be a nonlinear second-order differential equation in which is a rational function of and , and is locally analytic in , that is, analytic except for isolated singularities in . In general the singularities of the solutions are movable in the sense that their location depends on the constants of integration associated with the initial or boundary conditions. An equation is said to have the Painlevé property if all its solutions are free from movable branch points; the solutions may have movable poles or movable isolated essential singularities (§1.10(iii)), however.

There are fifty equations with the Painlevé property. They are distinct modulo Möbius (bilinear) transformations

in which , , , , and are locally analytic functions. The fifty equations can be reduced to linear equations, solved in terms of elliptic functions (Chapters 22 and 23), or reduced to one of .

For arbitrary values of the parameters , , , and , the general solutions of  are transcendental, that is, they cannot be expressed in closed-form elementary functions. However, for special values of the parameters, equations  have special solutions in terms of elementary functions, or special functions defined elsewhere in the DLMF.

## §32.2(ii) Renormalizations

If in , then set and , without loss of generality, by rescaling and if necessary. If and in , then set and , without loss of generality. Lastly, if and , then set and , without loss of generality.

If in , then set , without loss of generality.

## §32.2(iii) Alternative Forms

In , if , , and , then

In , if with and , then

32.2.11

When this is a nonlinear harmonic oscillator.

In , if with , then

32.2.12

See also Okamoto (1987c), McCoy et al. (1977), Bassom et al. (1992), Bassom et al. (1995), and Takasaki (2001).

## §32.2(iv) Elliptic Form

can be written in the form

where

See Fuchs (1907), Painlevé (1906), Gromak et al. (2002, §42); also Manin (1998).

## §32.2(v) Symmetric Forms

Let

where , , are constants, , , are functions of , with

32.2.16
32.2.17

Then satisfies  with

Next, let

where , , , are constants, , , , are functions of , with

32.2.20
32.2.21
32.2.22

Then satisfies  with

are obtained from  by a coalescence cascade:

32.2.24

For example, if in

32.2.26

then

thus in the limit as , satisfies  with .

If in

then as , satisfies  with , .

Lastly, if in

then as , satisfies  with , , , , .