32 Painlevé Transcendents Properties 32.2 Differential Equations 32.4 Isomonodromy Problems

§32.3 Graphics

ⓘ

Keywords:: Painlevé equations, Painlevé transcendents, graphs, graphs of solutions
Permalink:: http://dlmf.nist.gov/32.3
See also:: Annotations for Ch.32

§32.3(i) First Painlevé Equation

ⓘ

Notes:: Figures 32.3.1–32.3.4 were produced at NIST.
Permalink:: http://dlmf.nist.gov/32.3.i
See also:: Annotations for §32.3 and Ch.32

Plots of solutions $w_{k}(x)$ of $\mbox{P}_{\mbox{\scriptsize I}}$ with $w_{k}(0)=0$ and $w_{k}^{\prime}(0)=k$ for various values of $k$ , and the parabola $6w^{2}+x=0$ . For analytical explanation see §32.11(i).

See accompanying text — Figure 32.3.1: $w_{k}(x)$ for $-12\leq x\leq 1.33$ and $k=0.5$ , $0.75$ , $1$ , $1.25$ , and the parabola $6w^{2}+x=0$ , shown in black. Magnify

§32.3(ii) Second Painlevé Equation with $\alpha=0$

ⓘ

Notes:: Figures 32.3.5 and 32.3.6 were produced at NIST. See also Miles (1978, 1980) and Rosales (1978).
Permalink:: http://dlmf.nist.gov/32.3.ii
See also:: Annotations for §32.3 and Ch.32

Here $w_{k}(x)$ is the solution of $\mbox{P}_{\mbox{\scriptsize II}}$ with $\alpha=0$ and such that

32.3.1		$w_{k}(x)\sim k\operatorname{Ai}\left(x\right),$
$x\to+\infty$ ;
ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\sim$ : asymptotic equality, $x$ : real and $k$ : real Permalink: http://dlmf.nist.gov/32.3.E1 Encodings: TeX, pMML, png See also: Annotations for §32.3(ii), §32.3 and Ch.32

compare §32.11(ii).

§32.3(iii) Fourth Painlevé Equation with $\beta=0$

ⓘ

Keywords:: Painlevé equations, Painlevé transcendents, graphs, graphs of solutions
Notes:: Figures 32.3.7–32.3.10 were produced at NIST. See also Bassom et al. (1993).
Permalink:: http://dlmf.nist.gov/32.3.iii
See also:: Annotations for §32.3 and Ch.32

Here $u=u_{k}(x;\nu)$ is the solution of

32.3.2		$\frac{{\mathrm{d}}^{2}u}{{\mathrm{d}x}^{2}}=3u^{5}+2xu^{3}+\left(\tfrac{1}{4}x% ^{2}-\nu-\tfrac{1}{2}\right)u,$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $x$ : real, $\nu$ : parameter and $u_{k}(x;\nu)$ : solution Referenced by: §32.3(iii) Permalink: http://dlmf.nist.gov/32.3.E2 Encodings: TeX, pMML, png See also: Annotations for §32.3(iii), §32.3 and Ch.32

such that

32.3.3		$u\sim kU\left(-\nu-\tfrac{1}{2},x\right),$
$x\to+\infty$ .
ⓘ Symbols: $\sim$ : asymptotic equality, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $x$ : real, $k$ : real, $\nu$ : parameter and $u_{k}(x;\nu)$ : solution Permalink: http://dlmf.nist.gov/32.3.E3 Encodings: TeX, pMML, png See also: Annotations for §32.3(iii), §32.3 and Ch.32

The corresponding solution of $\mbox{P}_{\mbox{\scriptsize IV}}$ is given by

32.3.4		$w(x)=2\sqrt{2}u_{k}^{2}(\sqrt{2}x,\nu),$
ⓘ Symbols: $x$ : real, $k$ : real, $\nu$ : parameter and $u_{k}(x;\nu)$ : solution Permalink: http://dlmf.nist.gov/32.3.E4 Encodings: TeX, pMML, png See also: Annotations for §32.3(iii), §32.3 and Ch.32

with $\beta=0$ , $\alpha=2\nu+1$ , and

32.3.5		$w(x)\sim 2\sqrt{2}k^{2}{U}^{2}\left(-\nu-\tfrac{1}{2},\sqrt{2}x\right),$
$x\to+\infty$ ;
ⓘ Symbols: $\sim$ : asymptotic equality, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $x$ : real, $k$ : real and $\nu$ : parameter Permalink: http://dlmf.nist.gov/32.3.E5 Encodings: TeX, pMML, png See also: Annotations for §32.3(iii), §32.3 and Ch.32

compare (32.2.11) and §32.11(v). If we set $\ifrac{{\mathrm{d}}^{2}u}{{\mathrm{d}x}^{2}}=0$ in (32.3.2) and solve for $u$ , then

32.3.6		$u^{2}=-\tfrac{1}{3}x\pm\tfrac{1}{6}\sqrt{x^{2}+12\nu+6}.$
ⓘ Symbols: $x$ : real, $\nu$ : parameter and $u_{k}(x;\nu)$ : solution Permalink: http://dlmf.nist.gov/32.3.E6 Encodings: TeX, pMML, png See also: Annotations for §32.3(iii), §32.3 and Ch.32

32.2 Differential Equations 32.4 Isomonodromy Problems

NIST

Site Privacy Accessibility Privacy Program Copyrights Vulnerability Disclosure No Fear Act Policy FOIA Environmental Policy Scientific Integrity Information Quality Standards Commerce.gov Science.gov USA.gov

§32.3 Graphics

Contents

§32.3(i) First Painlevé Equation

§32.3(ii) Second Painlevé Equation with α=0

§32.3(iii) Fourth Painlevé Equation with β=0

§32.3(ii) Second Painlevé Equation with $\alpha=0$

§32.3(iii) Fourth Painlevé Equation with $\beta=0$