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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

Contents

§32.3(i) First Painlevé Equation
§32.3(ii) Second Painlevé Equation with $\alpha=0$
§32.3(iii) Fourth Painlevé Equation with $\beta=0$

§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 , 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

, 0.75, 1, 1.25, and the parabola $6w^{2}+x=0$ , shown in black. Magnify

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Symbols:: : real and : real
Referenced by:: §32.11(i), §32.3(i)
Permalink:: http://dlmf.nist.gov/32.3.F1
Encodings:: pdf, png
See also:: Annotations for §32.3(i), §32.3 and Ch.32

See accompanying text

Figure 32.3.2: $w_{k}(x)$ for $-12\leq x\leq 2.43$ and

, −0.25, 0, 1, 2, and the parabola $6w^{2}+x=0$ , shown in black. Magnify

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Symbols:: : real and : real
Permalink:: http://dlmf.nist.gov/32.3.F2
Encodings:: pdf, png
See also:: Annotations for §32.3(i), §32.3 and Ch.32

See accompanying text

Figure 32.3.3: $w_{k}(x)$ for $-12\leq x\leq 0.73$ and $k=1.85185\;3$ , 1.85185 5. The two graphs are indistinguishable when

exceeds −5.2, approximately. The parabola $6w^{2}+x=0$ is shown in black. Magnify

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Symbols:: : real and : real
Permalink:: http://dlmf.nist.gov/32.3.F3
Encodings:: pdf, png
See also:: Annotations for §32.3(i), §32.3 and Ch.32

See accompanying text

Figure 32.3.4: $w_{k}(x)$ for $-12\leq x\leq 2.3$ and $k=-0.45142\;7$ , −0.45142 8. The two graphs are indistinguishable when

exceeds −4.8, approximately. The parabola $6w^{2}+x=0$ is shown in black. Magnify

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Symbols:: : real and : real
Referenced by:: §32.11(i), §32.3(i)
Permalink:: http://dlmf.nist.gov/32.3.F4
Encodings:: pdf, png
See also:: Annotations for §32.3(i), §32.3 and Ch.32

§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 $w_{k}(x)\sim k\operatorname{Ai}\left(x\right),$ $x\to+\infty$ ;

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Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\sim$ : asymptotic equality, : real and : 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).

See accompanying text

Figure 32.3.5: $w_{k}(x)$ and $k\operatorname{Ai}\left(x\right)$ for $-10\leq x\leq 4$ with

. The two graphs are indistinguishable when

exceeds −0.4, approximately. Magnify

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Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, : real and : real
Referenced by:: §32.11(ii), §32.3(ii)
Permalink:: http://dlmf.nist.gov/32.3.F5
Encodings:: pdf, png
See also:: Annotations for §32.3(ii), §32.3 and Ch.32

See accompanying text

Figure 32.3.6: $w_{k}(x)$ for $-10\leq x\leq 4$ with

, 1.001. The two graphs are indistinguishable when

exceeds −2.8, approximately. The parabola $2w^{2}+x=0$ is shown in black. Magnify

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Symbols:: : real and : real
Referenced by:: §32.11(ii), §32.3(ii)
Permalink:: http://dlmf.nist.gov/32.3.F6
Encodings:: pdf, png
See also:: Annotations for §32.3(ii), §32.3 and Ch.32

§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 $\frac{{\mathrm{d}}^{2}u}{{\mathrm{d}x}^{2}}=3u^{5}+2xu^{3}+\left(\tfrac{1}{4}x% ^{2}-\nu-\tfrac{1}{2}\right)u,$

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Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of with respect to , : 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 $u\sim kU\left(-\nu-\tfrac{1}{2},x\right),$ $x\to+\infty$ .

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Symbols:: $\sim$ : asymptotic equality, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, : real, : 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 $w(x)=2\sqrt{2}u_{k}^{2}(\sqrt{2}x,\nu),$

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Symbols:: : real, : 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 $w(x)\sim 2\sqrt{2}k^{2}{U}^{2}\left(-\nu-\tfrac{1}{2},\sqrt{2}x\right),$ $x\to+\infty$ ;

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Symbols:: $\sim$ : asymptotic equality, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, : real, : 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 , then

32.3.6 $u^{2}=-\tfrac{1}{3}x\pm\tfrac{1}{6}\sqrt{x^{2}+12\nu+6}.$

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Symbols:: : 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

See accompanying text

Figure 32.3.7: $u_{k}(x;-\tfrac{1}{2})$ for $-12\leq x\leq 4$ with $k=0.33554\;691$ , 0.33554 692. The two graphs are indistinguishable when

exceeds −5.0, approximately. The parabolas $u^{2}+\tfrac{1}{2}x=0$ , $u^{2}+\tfrac{1}{6}x=0$ are shown in black and green, respectively. Magnify

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Symbols:: : real, : real and $u_{k}(x;\nu)$ : solution
Referenced by:: §32.11(v), §32.3(iii)
Permalink:: http://dlmf.nist.gov/32.3.F7
Encodings:: pdf, png
See also:: Annotations for §32.3(iii), §32.3 and Ch.32

See accompanying text

Figure 32.3.8: $u_{k}(x;\tfrac{1}{2})$ for $-12\leq x\leq 4$ with

, 0.47443. The two graphs are indistinguishable when

exceeds −2.2, approximately. The curves $u^{2}+\tfrac{1}{3}x\pm\tfrac{1}{6}\sqrt{x^{2}+12}=0$ are shown in green and black, respectively. Magnify

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Symbols:: : real, : real and $u_{k}(x;\nu)$ : solution
Permalink:: http://dlmf.nist.gov/32.3.F8
Encodings:: pdf, png
See also:: Annotations for §32.3(iii), §32.3 and Ch.32

See accompanying text

Figure 32.3.9: $u_{k}(x;\tfrac{3}{2})$ for $-12\leq x\leq 4$ with

, 0.38737. The two graphs are indistinguishable when

exceeds −1.0, approximately. The curves $u^{2}+\tfrac{1}{3}x\pm\tfrac{1}{6}\sqrt{x^{2}+24}=0$ are shown in green and black, respectively. Magnify

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Symbols:: : real, : real and $u_{k}(x;\nu)$ : solution
Permalink:: http://dlmf.nist.gov/32.3.F9
Encodings:: pdf, png
See also:: Annotations for §32.3(iii), §32.3 and Ch.32

See accompanying text

Figure 32.3.10: $u_{k}(x;\tfrac{5}{2})$ for $-12\leq x\leq 4$ with $k=0.24499\;2$ , 0.24499 3. The two graphs are indistinguishable when

exceeds −0.6, approximately. The curves $u^{2}+\tfrac{1}{3}x\pm\tfrac{1}{6}\sqrt{x^{2}+36}=0$ are shown in green and black, respectively. Magnify

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Symbols:: : real, : real and $u_{k}(x;\nu)$ : solution
Referenced by:: §32.11(v), §32.3(iii)
Permalink:: http://dlmf.nist.gov/32.3.F10
Encodings:: pdf, png
See also:: Annotations for §32.3(iii), §32.3 and Ch.32

32.2 Differential Equations 32.4 Isomonodromy Problems

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