§32.7 Bäcklund Transformations

With the exception of ${\text{P}}_{\text{I}}$, 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.

Let $w=w(z;\alpha )$ be a solution of ${\text{P}}_{\text{II}}$. Then the transformations

and

furnish solutions of ${\text{P}}_{\text{II}}$, provided that $\alpha \ne \mp \frac{1}{2}$. ${\text{P}}_{\text{II}}$ also has the special transformation

or equivalently,

with $\zeta =-2^{1/3}z$ and $\epsilon =\pm 1$, where $W(\zeta ;\frac{1}{2}\epsilon )$ satisfies ${\text{P}}_{\text{II}}$ with $z=\zeta$, $\alpha =\frac{1}{2}\epsilon$, and $w(z;0)$ satisfies ${\text{P}}_{\text{II}}$ with $\alpha =0$.

The solutions $w_{\alpha }=w(z;\alpha )$, $w_{\alpha \pm 1}=w(z;\alpha \pm 1)$, satisfy the nonlinear recurrence relation

See Fokas et al. (1993).

Let $w_j=w(z;{\alpha }_j,{\beta }_j,{\gamma }_j,{\delta }_j)$, $j=0,1,2$, be solutions of ${\text{P}}_{\text{III}}$ with

Then

Next, let $W_j=W(z;{\alpha }_j,{\beta }_j,1,-1)$, $j=0,1,2,3,4$, be solutions of ${\text{P}}_{\text{III}}$ with

Then

See Milne et al. (1997).

If $\gamma =0$ and $\alpha \delta \ne 0$, then set $\alpha =1$ and $\delta =-1$, without loss of generality. Let $u_j=w(z;1,{\beta }_j,0,-1)$, $j=0,5,6$, be solutions of ${\text{P}}_{\text{III}}$ with

Then

Similar results hold for ${\text{P}}_{\text{III}}$ with $\delta =0$ and $\beta \gamma \ne 0$.

Furthermore,

Let $w_0=w(z;{\alpha }_0,{\beta }_0)$ and $w_j^{\pm }=w(z;{\alpha }_j^{\pm },{\beta }_j^{\pm })$, $j=1,2,3,4$, be solutions of ${\text{P}}_{\text{IV}}$ 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).

Let $w_j(z_j)=w(z_j;{\alpha }_j,{\beta }_j,{\gamma }_j,{\delta }_j)$, $j=0,1,2$, be solutions of ${\text{P}}_{\text{V}}$ with

Then

Let $W_0=W(z;{\alpha }_0,{\beta }_0,{\gamma }_0,-\frac{1}{2})$ and $W_1=W(z;{\alpha }_1,{\beta }_1,{\gamma }_1,-\frac{1}{2})$ be solutions of ${\text{P}}_{\text{V}}$, where

and ${\epsilon }_j=\pm 1$, $j=1,2,3$, independently. Also let

and assume $\mathrm{\Phi }\ne 0$. Then

provided that the numerator on the right-hand side does not vanish. Again, since ${\epsilon }_j=\pm 1$, $j=1,2,3$, independently, there are eight distinct transformations of type ${𝒯}_{{\epsilon }_1,{\epsilon }_2,{\epsilon }_3}$.

Let $w=w(z;\alpha ,\beta ,1,-1)$ be a solution of ${\text{P}}_{\text{III}}$ and

with $\epsilon =\pm 1$. Then

satisfies ${\text{P}}_{\text{V}}$ with

Let $w_j(z_j)=w_j(z_j;{\alpha }_j,{\beta }_j,{\gamma }_j,{\delta }_j)$, $j=0,1,2,3$, be solutions of ${\text{P}}_{\text{VI}}$ with

Then

The transformations ${𝒮}_j$, for $j=1,2,3$, generate a group of order 24. See Iwasaki et al. (1991, p. 127).

Let $w(z;\alpha ,\beta ,\gamma ,\delta )$ and $W(z;A,B,C,D)$ be solutions of ${\text{P}}_{\text{VI}}$ with

and

for $j=0,1,2,\mathrm{\infty }$, where

Then

${\text{P}}_{\text{VI}}$ also has quadratic and quartic transformations. Let $w=w(z;\alpha ,\beta ,\gamma ,\delta )$ be a solution of ${\text{P}}_{\text{VI}}$. The quadratic transformation

transforms ${\text{P}}_{\text{VI}}$ with $\alpha =-\beta$ and $\gamma =\frac{1}{2}-\delta$ to ${\text{P}}_{\text{VI}}$ with $({\alpha }_1,{\beta }_1,{\gamma }_1,{\delta }_1)=(4\alpha ,-4\gamma ,0,\frac{1}{2})$. The quartic transformation

transforms ${\text{P}}_{\text{VI}}$ with $\alpha =-\beta =\gamma =\frac{1}{2}-\delta$ to ${\text{P}}_{\text{VI}}$ with $({\alpha }_2,{\beta }_2,{\gamma }_2,{\delta }_2)=(16\alpha ,0,0,\frac{1}{2})$. Also,

transforms ${\text{P}}_{\text{VI}}$ with $\alpha =\beta =0$ and $\gamma =\frac{1}{2}-\delta$ to ${\text{P}}_{\text{VI}}$ with ${\alpha }_3={\beta }_3$ and ${\gamma }_3=\frac{1}{2}-{\delta }_3$.

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