32 Painlevé Transcendents Properties 32.2 Differential Equations 32.4 Isomonodromy Problems

§32.3 Graphics

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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

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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 $\alpha=0$

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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 $\beta=0$

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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