# §32.11(i) First Painlevé Equation

There are solutions of (32.2.1) such that

 32.11.1 $w(x)=-\sqrt{\tfrac{1}{6}|x|}+d|x|^{-1/8}\mathop{\sin\/}\nolimits\!\left(\phi(x% )-\theta_{0}\right)+\mathop{o\/}\nolimits\!\left(|x|^{-1/8}\right),$ $x\to-\infty$,

where

 32.11.2 $\phi(x)=(24)^{1/4}\left(\tfrac{4}{5}|x|^{5/4}-\tfrac{5}{8}d^{2}\mathop{\ln\/}% \nolimits|x|\right),$

and $d$ and $\theta_{0}$ are constants.

There are also solutions of (32.2.1) such that

 32.11.3 $w(x)\sim\sqrt{\tfrac{1}{6}|x|},$ $x\to-\infty$. Symbols: $\sim$: asymptotic equality and $x$: real Permalink: http://dlmf.nist.gov/32.11.E3 Encodings: TeX, pMML, png

Next, for given initial conditions $w(0)=0$ and $w^{\prime}(0)=k$, with $k$ real, $w(x)$ has at least one pole on the real axis. There are two special values of $k$, $k_{1}$ and $k_{2}$, with the properties $-0.45142\;8, $1.85185\;3, and such that:

1. (a)

If $k, then $w(x)>0$ for $x_{0}, where $x_{0}$ is the first pole on the negative real axis.

2. (b)

If $k_{1}, then $w(x)$ oscillates about, and is asymptotic to, $-\sqrt{\tfrac{1}{6}|x|}$ as $x\to-\infty$.

3. (c)

If $k_{2}, then $w(x)$ changes sign once, from positive to negative, as $x$ passes from $x_{0}$ to $0$.

For illustration see Figures 32.3.1 to 32.3.4, and for further information see Joshi and Kitaev (2005), Joshi and Kruskal (1992), Kapaev (1988), Kapaev and Kitaev (1993), and Kitaev (1994).

# §32.11(ii) Second Painlevé Equation

Consider the special case of $\mbox{P}_{\mbox{\scriptsize II}}$ with $\alpha=0$:

 32.11.4 $w^{\prime\prime}=2w^{3}+xw,$ Symbols: $x$: real Referenced by: §32.11(ii), §32.11(iii) Permalink: http://dlmf.nist.gov/32.11.E4 Encodings: TeX, pMML, png

with boundary condition

 32.11.5 $w(x)\to 0,$ $x\to+\infty$. Symbols: $x$: real Referenced by: §32.11(ii) Permalink: http://dlmf.nist.gov/32.11.E5 Encodings: TeX, pMML, png

Any nontrivial real solution of (32.11.4) that satisfies (32.11.5) is asymptotic to $k\mathop{\mathrm{Ai}\/}\nolimits\!\left(x\right)$, for some nonzero real $k$, where $\mathop{\mathrm{Ai}\/}\nolimits$ denotes the Airy function (§9.2). Conversely, for any nonzero real $k$, there is a unique solution $w_{k}(x)$ of (32.11.4) that is asymptotic to $k\mathop{\mathrm{Ai}\/}\nolimits\!\left(x\right)$ as $x\to+\infty$.

If $|k|<1$, then $w_{k}(x)$ exists for all sufficiently large $|x|$ as $x\to-\infty$, and

 32.11.6 $w_{k}(x)=d|x|^{-1/4}\mathop{\sin\/}\nolimits\!\left(\phi(x)-\theta_{0}\right)+% \mathop{o\/}\nolimits\!\left(|x|^{-1/4}\right),$

where

 32.11.7 $\phi(x)=\tfrac{2}{3}|x|^{3/2}-\tfrac{3}{4}d^{2}\mathop{\ln\/}\nolimits|x|,$

and $d$ $(\neq 0)$, $\theta_{0}$ are real constants. Connection formulas for $d$ and $\theta_{0}$ are given by

 32.11.8 $d^{2}=-\pi^{-1}\mathop{\ln\/}\nolimits\!\left(1-k^{2}\right),$
 32.11.9 $\theta_{0}=\tfrac{3}{2}d^{2}\mathop{\ln\/}\nolimits 2+\mathop{\mathrm{ph}\/}% \nolimits\mathop{\Gamma\/}\nolimits\!\left(1-\tfrac{1}{2}id^{2}\right)+\tfrac{% 1}{4}\pi(1-2\mathop{\mathrm{sign}\/}\nolimits\!\left(k\right)),$

where $\mathop{\Gamma\/}\nolimits$ is the gamma function (§5.2(i)), and the branch of the $\mathop{\mathrm{ph}\/}\nolimits$ function is immaterial.

If $|k|=1$, then

 32.11.10 $w_{k}(x)\sim\mathop{\mathrm{sign}\/}\nolimits\!\left(k\right)\sqrt{\tfrac{1}{2% }|x|},$ $x\to-\infty$.

If $|k|>1$, then $w_{k}(x)$ has a pole at a finite point $x=c_{0}$, dependent on $k$, and

 32.11.11 $w_{k}(x)\sim\mathop{\mathrm{sign}\/}\nolimits\!\left(k\right)(x-c_{0})^{-1},$ $x\to c_{0}+$.

For illustration see Figures 32.3.5 and 32.3.6, and for further information see Ablowitz and Clarkson (1991), Bassom et al. (1998), Clarkson and McLeod (1988), Deift and Zhou (1995), Segur and Ablowitz (1981), and Suleĭmanov (1987). For numerical studies see Miles (1978, 1980) and Rosales (1978).

# §32.11(iii) Modified Second Painlevé Equation

Replacement of $w$ by $iw$ in (32.11.4) gives

 32.11.12 $w^{\prime\prime}=-2w^{3}+xw.$ Symbols: $x$: real Referenced by: §32.11(iii) Permalink: http://dlmf.nist.gov/32.11.E12 Encodings: TeX, pMML, png

Any nontrivial real solution of (32.11.12) satisfies

 32.11.13 $w(x)=d|x|^{-1/4}\mathop{\sin\/}\nolimits\!\left(\phi(x)-\chi\right)+\mathop{O% \/}\nolimits\!\left(|x|^{-5/4}\mathop{\ln\/}\nolimits|x|\right),$ $x\to-\infty$,

where

 32.11.14 $\phi(x)=\tfrac{2}{3}|x|^{3/2}+\tfrac{3}{4}d^{2}\mathop{\ln\/}\nolimits|x|,$

with $d$ $(\neq 0)$ and $\chi$ arbitrary real constants.

In the case when

 32.11.15 $\chi+\tfrac{3}{2}d^{2}\mathop{\ln\/}\nolimits 2-\tfrac{1}{4}\pi-\mathop{% \mathrm{ph}\/}\nolimits\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}id^{2}% \right)=n\pi,$

with $n\in\Integer$, we have

 32.11.16 $w(x)\sim k\mathop{\mathrm{Ai}\/}\nolimits\!\left(x\right),$ $x\to+\infty$,

where $k$ is a nonzero real constant. The connection formulas for $k$ are

 32.11.17 $d^{2}=\pi^{-1}\mathop{\ln\/}\nolimits\!\left(1+k^{2}\right),$ $\mathop{\mathrm{sign}\/}\nolimits\!\left(k\right)=(-1)^{n}$.

In the generic case

 32.11.18 $\chi+\tfrac{3}{2}d^{2}\mathop{\ln\/}\nolimits 2-\tfrac{1}{4}\pi-\mathop{% \mathrm{ph}\/}\nolimits\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}id^{2}% \right)\neq n\pi,$

we have

 32.11.19 $w(x)=\sigma\sqrt{\tfrac{1}{2}x}+\sigma\rho(2x)^{-1/4}\mathop{\cos\/}\nolimits% \!\left(\psi(x)+\theta\right)+\mathop{O\/}\nolimits\!\left(x^{-1}\right),$ $x\to+\infty$,

where $\sigma$, $\rho$ $(>0)$, and $\theta$ are real constants, and

 32.11.20 $\psi(x)=\tfrac{2}{3}\sqrt{2}x^{3/2}-\tfrac{3}{2}\rho^{2}\mathop{\ln\/}% \nolimits x.$

The connection formulas for $\sigma$, $\rho$, and $\theta$ are

 32.11.21 $\sigma=-\mathop{\mathrm{sign}\/}\nolimits\!\left(\imagpart{s}\right),$
 32.11.22 $\rho^{2}=\pi^{-1}\mathop{\ln\/}\nolimits\!\left((1+|s|^{2})/|2\imagpart{s}|% \right),$
 32.11.23 $\theta=-\tfrac{3}{4}\pi-\tfrac{7}{2}\rho^{2}\mathop{\ln\/}\nolimits{2}+\mathop% {\mathrm{ph}\/}\nolimits\!\left(1+s^{2}\right)+\mathop{\mathrm{ph}\/}\nolimits% \mathop{\Gamma\/}\nolimits\!\left(i\rho^{2}\right),$

where

 32.11.24 $s=\left(\mathop{\exp\/}\nolimits\!\left(\pi d^{2}\right)-1\right)^{1/2}\*% \mathop{\exp\/}\nolimits\!\left(i\left(\tfrac{3}{2}d^{2}\mathop{\ln\/}% \nolimits 2-\tfrac{1}{4}\pi+\chi-\mathop{\mathrm{ph}\/}\nolimits\mathop{\Gamma% \/}\nolimits\!\left(\tfrac{1}{2}id^{2}\right)\right)\right).$

# §32.11(iv) Third Painlevé Equation

For $\mbox{P}_{\mbox{\scriptsize III}}$, with $\alpha=-\beta=2\nu$ $(\in\Real)$ and $\gamma=-\delta=1$,

 32.11.25 $w(x)-1\sim-\lambda\mathop{\Gamma\/}\nolimits\!\left(\nu+\tfrac{1}{2}\right)2^{% -2\nu}x^{-\nu-(1/2)}e^{-2x},$ $x\to+\infty$,

where $\lambda$ is an arbitrary constant such that $-1/\pi<\lambda<1/\pi$, and

 32.11.26 $w(x)\sim Bx^{\sigma},$ $x\to 0$, Symbols: $\sim$: asymptotic equality, $x$: real, $B$: constant and $\sigma$: constant Referenced by: §32.11(iv) Permalink: http://dlmf.nist.gov/32.11.E26 Encodings: TeX, pMML, png

where $B$ and $\sigma$ are arbitrary constants such that $B\neq 0$ and $|\realpart{\sigma}|<1$. The connection formulas relating (32.11.25) and (32.11.26) are

 32.11.27 $\sigma=(2/\pi)\mathop{\mathrm{arcsin}\/}\nolimits\!\left(\pi\lambda\right),$
 32.11.28 $B=2^{-2\sigma}\frac{{\mathop{\Gamma\/}\nolimits^{2}}\!\left(\tfrac{1}{2}(1-% \sigma)\right)\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}(1+\sigma)+\nu% \right)}{{\mathop{\Gamma\/}\nolimits^{2}}\!\left(\tfrac{1}{2}(1+\sigma)\right)% \mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}(1-\sigma)+\nu\right)}.$

See also Abdullaev (1985), Novokshënov (1985), Its and Novokshënov (1986), Kitaev (1987), Bobenko (1991), Bobenko and Its (1995), Tracy and Widom (1997), and Kitaev and Vartanian (2004).

# §32.11(v) Fourth Painlevé Equation

Consider $\mbox{P}_{\mbox{\scriptsize IV}}$ with $\alpha=2\nu+1$ $(\in\Real)$ and $\beta=0$, that is,

 32.11.29 $w^{\prime\prime}=\frac{(w^{\prime})^{2}}{2w}+\frac{3}{2}w^{3}+4xw^{2}+2(x^{2}-% 2\nu-1)w,$ Symbols: $x$: real and $\nu$: parameter Referenced by: §32.11(v) Permalink: http://dlmf.nist.gov/32.11.E29 Encodings: TeX, pMML, png

and with boundary condition

 32.11.30 $w(x)\to 0,$ $x\to+\infty$. Symbols: $x$: real Referenced by: §32.11(v) Permalink: http://dlmf.nist.gov/32.11.E30 Encodings: TeX, pMML, png

Any nontrivial solution of (32.11.29) that satisfies (32.11.30) is asymptotic to $h{\mathop{U\/}\nolimits^{2}}\!\left(-\nu-\frac{1}{2},\sqrt{2}x\right)$ as $x\to+\infty$, where $h$ $(\neq 0)$ is a constant. Conversely, for any $h$ $(\neq 0)$ there is a unique solution $w_{h}(x)$ of (32.11.29) that is asymptotic to $h{\mathop{U\/}\nolimits^{2}}\!\left(-\nu-\frac{1}{2},\sqrt{2}x\right)$ as $x\to+\infty$. Here $\mathop{U\/}\nolimits$ denotes the parabolic cylinder function (§12.2).

Now suppose $x\to-\infty$. If $0\leq h, where

 32.11.31 $h^{*}=\ifrac{1}{\left(\pi^{1/2}\mathop{\Gamma\/}\nolimits\!\left(\nu+1\right)% \right)},$

then $w_{h}(x)$ has no poles on the real axis. Furthermore, if $\nu=n=0,1,2,\dots$, then

 32.11.32 $w_{h}(x)\sim h2^{n}x^{2n}\mathop{\exp\/}\nolimits\!\left(-x^{2}\right),$ $x\to-\infty$.

Alternatively, if $\nu$ is not zero or a positive integer, then

 32.11.33 $w_{h}(x)=-\tfrac{2}{3}x+\tfrac{4}{3}d\sqrt{3}\mathop{\sin\/}\nolimits\!\left(% \phi(x)-\theta_{0}\right)+\mathop{O\/}\nolimits\!\left(x^{-1}\right),$ $x\to-\infty$,

where

 32.11.34 $\phi(x)=\tfrac{1}{3}\sqrt{3}x^{2}-\tfrac{4}{3}d^{2}\sqrt{3}\mathop{\ln\/}% \nolimits\!\left(\sqrt{2}|x|\right),$

and $d$ $(>0)$ and $\theta_{0}$ are real constants. Connection formulas for $d$ and $\theta_{0}$ are given by

 32.11.35 $\displaystyle d^{2}$ $\displaystyle=-\tfrac{1}{4}\sqrt{3}\pi^{-1}\mathop{\ln\/}\nolimits\!\left(1-|% \mu|^{2}\right),$ 32.11.36 $\displaystyle\theta_{0}$ $\displaystyle=\tfrac{1}{3}d^{2}\sqrt{3}\mathop{\ln\/}\nolimits 3+\tfrac{2}{3}% \pi\nu+\tfrac{7}{12}\pi+\mathop{\mathrm{ph}\/}\nolimits\mu+\mathop{\mathrm{ph}% \/}\nolimits\mathop{\Gamma\/}\nolimits\!\left(-\tfrac{2}{3}i\sqrt{3}d^{2}% \right),$

where

 32.11.37 $\mu=1+\left(\ifrac{2ih\pi^{3/2}\mathop{\exp\/}\nolimits\!\left(-i\pi\nu\right)% }{\mathop{\Gamma\/}\nolimits\!\left(-\nu\right)}\right),$

and the branch of the $\mathop{\mathrm{ph}\/}\nolimits$ function is immaterial.

Next if $h=h^{*}$, then

 32.11.38 $w_{h^{*}}(x)\sim-2x,$ $x\to-\infty$, Symbols: $\sim$: asymptotic equality and $x$: real Permalink: http://dlmf.nist.gov/32.11.E38 Encodings: TeX, pMML, png

and $w_{h^{*}}(x)$ has no poles on the real axis.

Lastly if $h>h^{*}$, then $w_{h}(x)$ has a simple pole on the real axis, whose location is dependent on $h$.

For illustration see Figures 32.3.732.3.10. In terms of the parameter $k$ that is used in these figures $h=2^{3/2}k^{2}$.