# §32.7(i) Definition

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

# §32.7(ii) Second Painlevé Equation

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

 32.7.1 $\mathcal{S}:\enskip w(z;-\alpha)=-w,$ Symbols: $z$: real, $\alpha$: arbitrary constant and $\mathcal{S}$: transformation Referenced by: §32.10(ii), §32.8(ii) Permalink: http://dlmf.nist.gov/32.7.E1 Encodings: TeX, pMML, png

and

 32.7.2 $\mathcal{T}^{\pm}:\enskip w(z;\alpha\pm 1)=-w-\frac{2\alpha\pm 1}{2w^{2}\pm 2w% ^{\prime}+z},$ Symbols: $z$: real, $\alpha$: arbitrary constant and $\mathcal{T}^{\pm}$: transformation Referenced by: §32.10(ii), §32.8(ii) Permalink: http://dlmf.nist.gov/32.7.E2 Encodings: TeX, pMML, png

furnish solutions of $\mbox{P}_{\mbox{\scriptsize II}}$, provided that $\alpha\neq\mp\tfrac{1}{2}$. $\mbox{P}_{\mbox{\scriptsize II}}$ also has the special transformation

 32.7.3 $W(\zeta;\tfrac{1}{2}\varepsilon)=\frac{2^{-1/3}\varepsilon}{w(z;0)}\frac{d}{dz% }w(z;0),$

or equivalently,

 32.7.4 $w^{2}(z;0)=2^{-1/3}\left(W^{2}(\zeta;\tfrac{1}{2}\varepsilon)-\varepsilon\frac% {d}{d\zeta}W(\zeta;\tfrac{1}{2}\varepsilon)+\tfrac{1}{2}\zeta\right),$

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

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

 32.7.5 $\frac{\alpha+\tfrac{1}{2}}{w_{\alpha+1}+w_{\alpha}}+\frac{\alpha-\tfrac{1}{2}}% {w_{\alpha}+w_{\alpha-1}}+2w_{\alpha}^{2}+z=0.$ Symbols: $z$: real and $\alpha$: arbitrary constant Permalink: http://dlmf.nist.gov/32.7.E5 Encodings: TeX, pMML, png

See Fokas et al. (1993).

# §32.7(iii) Third Painlevé Equation

Let $w_{j}=w(z;\alpha_{j},\beta_{j},\gamma_{j},\delta_{j})$, $j=0,1,2$, be solutions of $\mbox{P}_{\mbox{\scriptsize III}}$ with

 32.7.6 $(\alpha_{1},\beta_{1},\gamma_{1},\delta_{1})=(-\alpha_{0},-\beta_{0},\gamma_{0% },\delta_{0}),$
 32.7.7 $(\alpha_{2},\beta_{2},\gamma_{2},\delta_{2})=(-\beta_{0},-\alpha_{0},-\delta_{% 0},-\gamma_{0}).$

Then

 32.7.8 $\displaystyle\mathcal{S}_{1}:\enskip w_{1}$ $\displaystyle=-w_{0},$ Symbols: $\mathcal{S}_{j}$: transformation Permalink: http://dlmf.nist.gov/32.7.E8 Encodings: TeX, pMML, png 32.7.9 $\displaystyle\mathcal{S}_{2}:\enskip w_{2}$ $\displaystyle=\ifrac{1}{w_{0}}.$ Symbols: $\mathcal{S}_{j}$: transformation Permalink: http://dlmf.nist.gov/32.7.E9 Encodings: TeX, pMML, png

Next, let $W_{j}=W(z;\alpha_{j},\beta_{j},1,-1)$, $j=0,1,2,3,4$, be solutions of $\mbox{P}_{\mbox{\scriptsize III}}$ with

 32.7.10 $\displaystyle\alpha_{1}$ $\displaystyle=\alpha_{3}=\alpha_{0}+2,$ $\displaystyle\alpha_{2}$ $\displaystyle=\alpha_{4}=\alpha_{0}-2,$ $\displaystyle\beta_{1}$ $\displaystyle=\beta_{2}=\beta_{0}+2,$ $\displaystyle\beta_{3}$ $\displaystyle=\beta_{4}=\beta_{0}-2.$ Symbols: $\alpha$: arbitrary constant and $\beta$: arbitrary constant Permalink: http://dlmf.nist.gov/32.7.E10 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png

Then

 32.7.11 $\displaystyle\mathcal{T}_{1}:\enskip W_{1}$ $\displaystyle=\frac{zW_{0}^{\prime}+zW_{0}^{2}-\beta W_{0}-W_{0}+z}{W_{0}(zW_{% 0}^{\prime}+zW_{0}^{2}+\alpha W_{0}+W_{0}+z)},$ 32.7.12 $\displaystyle\mathcal{T}_{2}:\enskip W_{2}$ $\displaystyle=-\frac{zW_{0}^{\prime}-zW_{0}^{2}-\beta W_{0}-W_{0}+z}{W_{0}(zW_% {0}^{\prime}-zW_{0}^{2}-\alpha W_{0}+W_{0}+z)},$ 32.7.13 $\displaystyle\mathcal{T}_{3}:\enskip W_{3}$ $\displaystyle=-\frac{zW_{0}^{\prime}+zW_{0}^{2}+\beta W_{0}-W_{0}-z}{W_{0}(zW_% {0}^{\prime}+zW_{0}^{2}+\alpha W_{0}+W_{0}-z)},$ 32.7.14 $\displaystyle\mathcal{T}_{4}:\enskip W_{4}$ $\displaystyle=\frac{zW_{0}^{\prime}-zW_{0}^{2}+\beta W_{0}-W_{0}-z}{W_{0}(zW_{% 0}^{\prime}-zW_{0}^{2}-\alpha W_{0}+W_{0}-z)}.$

See Milne et al. (1997).

If $\gamma=0$ and $\alpha\delta\neq 0$, 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 $\mbox{P}_{\mbox{\scriptsize III}}$ with

 32.7.15 $\displaystyle\beta_{5}$ $\displaystyle=\beta_{0}+2,$ $\displaystyle\beta_{6}$ $\displaystyle=\beta_{0}-2.$ Symbols: $\beta$: arbitrary constant Permalink: http://dlmf.nist.gov/32.7.E15 Encodings: TeX, TeX, pMML, pMML, png, png

Then

 32.7.16 $\displaystyle\mathcal{T}_{5}:\enskip u_{5}$ $\displaystyle=\ifrac{(zu_{0}^{\prime}+z-(\beta_{0}+1)u_{0})}{u_{0}^{2}},$ 32.7.17 $\displaystyle\mathcal{T}_{6}:\enskip u_{6}$ $\displaystyle=-\ifrac{(zu_{0}^{\prime}-z+(\beta_{0}-1)u_{0})}{u_{0}^{2}}.$

Similar results hold for $\mbox{P}_{\mbox{\scriptsize III}}$ with $\delta=0$ and $\beta\gamma\neq 0$.

Furthermore,

 32.7.18 $\displaystyle w(z;a,b,0,0)$ $\displaystyle=W^{2}(\zeta;0,0,a,b),$ $\displaystyle z$ $\displaystyle=\tfrac{1}{2}\zeta^{2}.$ Symbols: $z$: real and $W(\zeta,\varepsilon/2)$: transformation Permalink: http://dlmf.nist.gov/32.7.E18 Encodings: TeX, TeX, pMML, pMML, png, png

# §32.7(iv) Fourth Painlevé Equation

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 $\mbox{P}_{\mbox{\scriptsize IV}}$ with

 32.7.19 $\displaystyle\alpha_{1}^{\pm}$ $\displaystyle=\tfrac{1}{4}\left(2-2\alpha_{0}\pm 3\sqrt{-2\beta_{0}}\right),$ $\displaystyle\beta_{1}^{\pm}$ $\displaystyle=-\tfrac{1}{2}\left(1+\alpha_{0}\pm\tfrac{1}{2}\sqrt{-2\beta_{0}}% \right)^{2},$ $\displaystyle\alpha_{2}^{\pm}$ $\displaystyle=-\tfrac{1}{4}\left(2+2\alpha_{0}\pm 3\sqrt{-2\beta_{0}}\right),$ $\displaystyle\beta_{2}^{\pm}$ $\displaystyle=-\tfrac{1}{2}\left(1-\alpha_{0}\pm\tfrac{1}{2}\sqrt{-2\beta_{0}}% \right)^{2},$ $\displaystyle\alpha_{3}^{\pm}$ $\displaystyle=\tfrac{3}{2}-\tfrac{1}{2}\alpha_{0}\mp\tfrac{3}{4}\sqrt{-2\beta_% {0}},$ $\displaystyle\beta_{3}^{\pm}$ $\displaystyle=-\tfrac{1}{2}\left(1-\alpha_{0}\pm\tfrac{1}{2}\sqrt{-2\beta_{0}}% \right)^{2},$ $\displaystyle\alpha_{4}^{\pm}$ $\displaystyle=-\tfrac{3}{2}-\tfrac{1}{2}\alpha_{0}\mp\tfrac{3}{4}\sqrt{-2\beta% _{0}},$ $\displaystyle\beta_{4}^{\pm}$ $\displaystyle=-\tfrac{1}{2}\left(-1-\alpha_{0}\pm\tfrac{1}{2}\sqrt{-2\beta_{0}% }\right)^{2}.$ Symbols: $\alpha$: arbitrary constant and $\beta$: arbitrary constant Permalink: http://dlmf.nist.gov/32.7.E19 Encodings: TeX, TeX, TeX, TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png, png, png, png

Then

 32.7.20 $\displaystyle\mathcal{T}_{1}^{\pm}:\enskip w_{1}^{\pm}$ $\displaystyle=\frac{w_{0}^{\prime}-w_{0}^{2}-2zw_{0}\mp\sqrt{-2\beta_{0}}}{2w_% {0}},$ 32.7.21 $\displaystyle\mathcal{T}_{2}^{\pm}:\enskip w_{2}^{\pm}$ $\displaystyle=-\frac{w_{0}^{\prime}+w_{0}^{2}+2zw_{0}\mp\sqrt{-2\beta_{0}}}{2w% _{0}},$ 32.7.22 $\displaystyle\mathcal{T}_{3}^{\pm}:\enskip w_{3}^{\pm}$ $\displaystyle=w_{0}+\frac{2\left(1-\alpha_{0}\mp\tfrac{1}{2}\sqrt{-2\beta_{0}}% \right)w_{0}}{w_{0}^{\prime}\pm\sqrt{-2\beta_{0}}+2zw_{0}+w_{0}^{2}},$ 32.7.23 $\displaystyle\mathcal{T}_{4}^{\pm}:\enskip w_{4}^{\pm}$ $\displaystyle=w_{0}+\frac{2\left(1+\alpha_{0}\pm\tfrac{1}{2}\sqrt{-2\beta_{0}}% \right)w_{0}}{w_{0}^{\prime}\mp\sqrt{-2\beta_{0}}-2zw_{0}-w_{0}^{2}},$

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 $w_{j}(z_{j})=w(z_{j};\alpha_{j},\beta_{j},\gamma_{j},\delta_{j})$, $j=0,1,2$, be solutions of $\mbox{P}_{\mbox{\scriptsize V}}$ with

 32.7.24 $\displaystyle z_{1}$ $\displaystyle=-z_{0},$ $\displaystyle z_{2}$ $\displaystyle=z_{0},$ $\displaystyle(\alpha_{1},\beta_{1},\gamma_{1},\delta_{1})$ $\displaystyle=(\alpha_{0},\beta_{0},-\gamma_{0},\delta_{0}),$ $\displaystyle(\alpha_{2},\beta_{2},\gamma_{2},\delta_{2})$ $\displaystyle=(-\beta_{0},-\alpha_{0},-\gamma_{0},\delta_{0}).$

Then

 32.7.25 $\displaystyle\mathcal{S}_{1}:\enskip w_{1}(z_{1})$ $\displaystyle=w(z_{0}),$ Symbols: $z$: real and $\mathcal{S}_{j}$: transformation Permalink: http://dlmf.nist.gov/32.7.E25 Encodings: TeX, pMML, png 32.7.26 $\displaystyle\mathcal{S}_{2}:\enskip w_{2}(z_{2})$ $\displaystyle=\ifrac{1}{w(z_{0})}.$ Symbols: $z$: real and $\mathcal{S}_{j}$: transformation Permalink: http://dlmf.nist.gov/32.7.E26 Encodings: TeX, pMML, png

Let $W_{0}=W(z;\alpha_{0},\beta_{0},\gamma_{0},-\tfrac{1}{2})$ and $W_{1}=W(z;\alpha_{1},\beta_{1},\gamma_{1},-\tfrac{1}{2})$ be solutions of $\mbox{P}_{\mbox{\scriptsize V}}$, where

 32.7.27 $\displaystyle\alpha_{1}$ $\displaystyle=\tfrac{1}{8}\left(\gamma_{0}+\varepsilon_{1}\left(1-\varepsilon_% {3}\sqrt{-2\beta_{0}}-\varepsilon_{2}\sqrt{2\alpha_{0}}\right)\right)^{2},$ $\displaystyle\beta_{1}$ $\displaystyle=-\tfrac{1}{8}\left(\gamma_{0}-\varepsilon_{1}\left(1-\varepsilon% _{3}\sqrt{-2\beta_{0}}-\varepsilon_{2}\sqrt{2\alpha_{0}}\right)\right)^{2},$ $\displaystyle\gamma_{1}$ $\displaystyle=\varepsilon_{1}\left(\varepsilon_{3}\sqrt{-2\beta_{0}}-% \varepsilon_{2}\sqrt{2\alpha_{0}}\right),$

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

 32.7.28 $\Phi=zW_{0}^{\prime}-\varepsilon_{2}\sqrt{2\alpha_{0}}W_{0}^{2}+\varepsilon_{3% }\sqrt{-2\beta_{0}}+\left(\varepsilon_{2}\sqrt{2\alpha_{0}}-\varepsilon_{3}% \sqrt{-2\beta_{0}}+\varepsilon_{1}z\right)W_{0},$

and assume $\Phi\neq 0$. Then

 32.7.29 $\mathcal{T}_{\varepsilon_{1},\varepsilon_{2},\varepsilon_{3}}:\enskip W_{1}=% \ifrac{(\Phi-2\varepsilon_{1}zW_{0})}{\Phi},$

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

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

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

 32.7.30 $v=w^{\prime}-\varepsilon w^{2}+(\ifrac{(1-\varepsilon\alpha)w}{z}),$

with $\varepsilon=\pm 1$. Then

 32.7.31 $\displaystyle W(\zeta;\alpha_{0},\beta_{0},\gamma_{0},\delta_{0})$ $\displaystyle=\frac{v-1}{v+1},$ $\displaystyle z$ $\displaystyle=\sqrt{2\zeta},$

satisfies $\mbox{P}_{\mbox{\scriptsize V}}$ with

 32.7.32 $(\alpha_{0},\beta_{0},\gamma_{0},\delta_{0})={\left((\beta-\varepsilon\alpha+2% )^{2}/32,-(\beta+\varepsilon\alpha-2)^{2}/32,-\varepsilon,0\right)}.$

# §32.7(vii) Sixth Painlevé Equation

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 $\mbox{P}_{\mbox{\scriptsize VI}}$ with

 32.7.33 $\displaystyle z_{1}$ $\displaystyle=1/z_{0},$ Symbols: $z$: real Permalink: http://dlmf.nist.gov/32.7.E33 Encodings: TeX, pMML, png 32.7.34 $\displaystyle z_{2}$ $\displaystyle=1-z_{0},$ Symbols: $z$: real Permalink: http://dlmf.nist.gov/32.7.E34 Encodings: TeX, pMML, png 32.7.35 $\displaystyle z_{3}$ $\displaystyle=1/z_{0},$ Symbols: $z$: real Permalink: http://dlmf.nist.gov/32.7.E35 Encodings: TeX, pMML, png
 32.7.36 $(\alpha_{1},\beta_{1},\gamma_{1},\delta_{1})=(\alpha_{0},\beta_{0},-\delta_{0}% +\tfrac{1}{2},-\gamma_{0}+\tfrac{1}{2}),$
 32.7.37 $(\alpha_{2},\beta_{2},\gamma_{2},\delta_{2})=(\alpha_{0},-\gamma_{0},-\beta_{0% },\delta_{0}),$
 32.7.38 $(\alpha_{3},\beta_{3},\gamma_{3},\delta_{3})=(-\beta_{0},-\alpha_{0},\gamma_{0% },\delta_{0}).$

Then

 32.7.39 $\displaystyle\mathcal{S}_{1}:\enskip w_{1}(z_{1})$ $\displaystyle=w_{0}(z_{0})/z_{0},$ Symbols: $z$: real and $\mathcal{S}_{j}$: transformation Permalink: http://dlmf.nist.gov/32.7.E39 Encodings: TeX, pMML, png 32.7.40 $\displaystyle\mathcal{S}_{2}:\enskip w_{2}(z_{2})$ $\displaystyle=1-w_{0}(z_{0}),$ Symbols: $z$: real and $\mathcal{S}_{j}$: transformation Permalink: http://dlmf.nist.gov/32.7.E40 Encodings: TeX, pMML, png 32.7.41 $\displaystyle\mathcal{S}_{3}:\enskip w_{3}(z_{3})$ $\displaystyle=1/w_{0}(z_{0}).$ Symbols: $z$: real and $\mathcal{S}_{j}$: transformation Permalink: http://dlmf.nist.gov/32.7.E41 Encodings: TeX, pMML, png

The transformations $\mathcal{S}_{j}$, 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 $\mbox{P}_{\mbox{\scriptsize VI}}$ with

 32.7.42 $(\alpha,\beta,\gamma,\delta)=\left(\tfrac{1}{2}(\theta_{\infty}-1)^{2},-\tfrac% {1}{2}\theta_{0}^{2},\tfrac{1}{2}\theta_{1}^{2},\tfrac{1}{2}(1-\theta_{2}^{2})% \right),$
 32.7.43 $(A,B,C,D)=\left(\tfrac{1}{2}(\Theta_{\infty}-1)^{2},-\tfrac{1}{2}\Theta_{0}^{2% },\tfrac{1}{2}\Theta_{1}^{2},\tfrac{1}{2}(1-\Theta_{2}^{2})\right),$ Symbols: $\Theta_{j}$: transformation Permalink: http://dlmf.nist.gov/32.7.E43 Encodings: TeX, pMML, png

and

 32.7.44 $\theta_{j}=\Theta_{j}+\tfrac{1}{2}\sigma,$

for $j=0,1,2,\infty$, where

 32.7.45 $\sigma=\theta_{0}+\theta_{1}+\theta_{2}+\theta_{\infty}-1=1-(\Theta_{0}+\Theta% _{1}+\Theta_{2}+\Theta_{\infty}).$

Then

 32.7.46 $\frac{\sigma}{w-W}=\frac{z(z-1)W^{\prime}}{W(W-1)(W-z)}+\frac{\Theta_{0}}{W}+% \frac{\Theta_{1}}{W-1}+\frac{\Theta_{2}-1}{W-z}=\frac{z(z-1)w^{\prime}}{w(w-1)% (w-z)}+\frac{\theta_{0}}{w}+\frac{\theta_{1}}{w-1}+\frac{\theta_{2}-1}{w-z}.$

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

 32.7.47 $\displaystyle u_{1}(\zeta_{1})$ $\displaystyle=\frac{(1-w)(w-z)}{(1+\sqrt{z})^{2}w},$ $\displaystyle\zeta_{1}$ $\displaystyle=\left(\frac{1-\sqrt{z}}{1+\sqrt{z}}\right)^{2},$

transforms $\mbox{P}_{\mbox{\scriptsize VI}}$ with $\alpha=-\beta$ and $\gamma=\tfrac{1}{2}-\delta$ to $\mbox{P}_{\mbox{\scriptsize VI}}$ with $(\alpha_{1},\beta_{1},\gamma_{1},\delta_{1})=(4\alpha,-4\gamma,0,\tfrac{1}{2})$. The quartic transformation

 32.7.48 $\displaystyle u_{2}(\zeta_{2})$ $\displaystyle=\frac{(w^{2}-z)^{2}}{4w(w-1)(w-z)},$ $\displaystyle\zeta_{2}$ $\displaystyle=z,$

transforms $\mbox{P}_{\mbox{\scriptsize VI}}$ with $\alpha=-\beta=\gamma=\tfrac{1}{2}-\delta$ to $\mbox{P}_{\mbox{\scriptsize VI}}$ with $(\alpha_{2},\beta_{2},\gamma_{2},\delta_{2})=(16\alpha,0,0,\tfrac{1}{2})$. Also,

 32.7.49 $u_{3}(\zeta_{3})=\left(\frac{1-z^{1/4}}{1+z^{1/4}}\right)^{2}\left(\frac{\sqrt% {w}+z^{1/4}}{\sqrt{w}-z^{1/4}}\right)^{2},$
 32.7.50 $\zeta_{3}=\left(\frac{1-z^{1/4}}{1+z^{1/4}}\right)^{4},$ Symbols: $z$: real and $\zeta_{3}$: transformation Permalink: http://dlmf.nist.gov/32.7.E50 Encodings: TeX, pMML, png

transforms $\mbox{P}_{\mbox{\scriptsize VI}}$ with $\alpha=\beta=0$ and $\gamma=\tfrac{1}{2}-\delta$ to $\mbox{P}_{\mbox{\scriptsize VI}}$ with $\alpha_{3}=\beta_{3}$ and $\gamma_{3}=\tfrac{1}{2}-\delta_{3}$.

# §32.7(viii) Affine Weyl Groups

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