# §32.7 Bäcklund Transformations

## §32.7(i) Definition

With the exception of , a Bäcklund transformation relates a Painlevé transcendent of one type either to another of the same type but with different values of the parameters, or to another type.

## §32.7(ii) Second Painlevé Equation

Let be a solution of . Then the transformations

32.7.1

and

32.7.2

furnish solutions of , provided that .  also has the special transformation

or equivalently,

with and , where satisfies  with , , and satisfies  with .

The solutions , , satisfy the nonlinear recurrence relation

32.7.5

See Fokas et al. (1993).

## §32.7(iii) Third Painlevé Equation

Let , , be solutions of  with

Then

32.7.8
32.7.9

If and , then set and , without loss of generality. Let , , be solutions of  with

32.7.15

Then

32.7.16
32.7.17

Similar results hold for  with and .

Furthermore,

32.7.18

## §32.7(iv) Fourth Painlevé Equation

Let and , , be solutions of  with

Then

valid when the denominators are nonzero, and where the upper signs or the lower signs are taken throughout each transformation. See Bassom et al. (1995).

## §32.7(v) Fifth Painlevé Equation

Let , , be solutions of  with

Then

32.7.25
32.7.26

Let and be solutions of , where

and , , independently. Also let

and assume . Then

32.7.29

provided that the numerator on the right-hand side does not vanish. Again, since , , independently, there are eight distinct transformations of type .

## §32.7(vi) Relationship Between the Third and Fifth Painlevé Equations

Let be a solution of  and

32.7.30

with . Then

satisfies  with

## §32.7(vii) Sixth Painlevé Equation

Let , , be solutions of  with

32.7.33
32.7.34
32.7.35

Then

32.7.39
32.7.40
32.7.41

The transformations , for , generate a group of order 24. See Iwasaki et al. (1991, p. 127).

Let and be solutions of  with

32.7.43

and

32.7.44

for , where

32.7.45

Then

also has quadratic and quartic transformations. Let be a solution of . The quadratic transformation

transforms  with and to  with . The quartic transformation

transforms  with to  with . Also,

32.7.49
32.7.50

transforms  with and to  with and .

## §32.7(viii) Affine Weyl Groups

See Okamoto (1986, 1987a, 1987b, 1987c), Sakai (2001), Umemura (2000).