§32.3 Graphics

Keywords:: Painlevé equations, Painlevé transcendents
Permalink:: http://dlmf.nist.gov/32.3

§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

Notes:: Figures 32.3.1–32.3.4 were produced at NIST.
Permalink:: http://dlmf.nist.gov/32.3.i

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).

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.

Symbols:: $x$ : real and $k$ : real
Referenced by:: §32.11(i), §32.3(i)
Permalink:: http://dlmf.nist.gov/32.3.F1
Encodings:: pdf, png

Figure 32.3.2: $w_{k}(x)$ for $-12\leq x\leq 2.43$ and $k=-0.5$ , −0.25, 0, 1, 2, and the parabola $6w^{2}+x=0$ , shown in black.

Symbols:: $x$ : real and $k$ : real
Permalink:: http://dlmf.nist.gov/32.3.F2
Encodings:: pdf, png

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

Symbols:: $x$ : real and $k$ : real
Permalink:: http://dlmf.nist.gov/32.3.F3
Encodings:: pdf, png

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

Symbols:: $x$ : real and $k$ : real
Referenced by:: §32.11(i), §32.3(i)
Permalink:: http://dlmf.nist.gov/32.3.F4
Encodings:: pdf, png

§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

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

32.3.1 $w_{k}(x)\sim k\mathop{\mathrm{Ai}\/}\nolimits\!\left(x\right),$ $x\to+\infty$ ;

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\sim$ : asymptotic equality, $x$ : real and $k$ : real
Permalink:: http://dlmf.nist.gov/32.3.E1
Encodings:: TeX, pMML, png

compare §32.11(ii).

Figure 32.3.5: $w_{k}(x)$ and $k\mathop{\mathrm{Ai}\/}\nolimits\!\left(x\right)$ for $-10\leq x\leq 4$ with $k=0.5$ . The two graphs are indistinguishable when $x$ exceeds −0.4, approximately.

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $x$ : real and $k$ : real
Referenced by:: §32.11(ii), §32.3(ii)
Permalink:: http://dlmf.nist.gov/32.3.F5
Encodings:: pdf, png

Figure 32.3.6: $w_{k}(x)$ for $-10\leq x\leq 4$ with $k=0.999$ , 1.001. The two graphs are indistinguishable when $x$ exceeds −2.8, approximately. The parabola $2w^{2}+x=0$ is shown in black.

Symbols:: $x$ : real and $k$ : real
Referenced by:: §32.11(ii), §32.3(ii)
Permalink:: http://dlmf.nist.gov/32.3.F6
Encodings:: pdf, png

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

Notes:: Figures 32.3.7–32.3.10 were produced at NIST. See also Bassom et al. (1993).
Keywords:: Painlevé equations, Painlevé transcendents
Permalink:: http://dlmf.nist.gov/32.3.iii

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

32.3.2 $\frac{{d}^{2}u}{{dx}^{2}}=3u^{5}+2xu^{3}+\left(\tfrac{1}{4}x^{2}-\nu-\tfrac{1}{2}\right)u,$

Symbols:: $\frac{df}{dx}$ : 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

such that

32.3.3 $u\sim k\mathop{U\/}\nolimits\!\left(-\nu-\tfrac{1}{2},x\right),$ $x\to+\infty$ .

Symbols:: $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, $\sim$ : asymptotic equality, $x$ : real, $k$ : real, $\nu$ : parameter and $u_{k}(x;\nu)$ : solution
Permalink:: http://dlmf.nist.gov/32.3.E3
Encodings:: TeX, pMML, png

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

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

32.3.5 $w(x)\sim 2\sqrt{2}k^{2}{\mathop{U\/}\nolimits^{{2}}}\!\left(-\nu-\tfrac{1}{2},\sqrt{2}x\right),$ $x\to+\infty$ ;

Symbols:: $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, $\sim$ : asymptotic equality, $x$ : real, $k$ : real and $\nu$ : parameter
Permalink:: http://dlmf.nist.gov/32.3.E5
Encodings:: TeX, pMML, png

compare (32.2.11) and §32.11(v). If we set $\ifrac{{d}^{2}u}{{dx}^{2}}=0$ 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

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 $x$ 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.

Symbols:: $x$ : real, $k$ : 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

Figure 32.3.8: $u_{k}(x;\tfrac{1}{2})$ for $-12\leq x\leq 4$ with $k=0.47442$ , 0.47443. The two graphs are indistinguishable when $x$ 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.

Symbols:: $x$ : real, $k$ : real and $u_{k}(x;\nu)$ : solution
Permalink:: http://dlmf.nist.gov/32.3.F8
Encodings:: pdf, png

Figure 32.3.9: $u_{k}(x;\tfrac{3}{2})$ for $-12\leq x\leq 4$ with $k=0.38736$ , 0.38737. The two graphs are indistinguishable when $x$ 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.

Symbols:: $x$ : real, $k$ : real and $u_{k}(x;\nu)$ : solution
Permalink:: http://dlmf.nist.gov/32.3.F9
Encodings:: pdf, png

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 $x$ 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.

Symbols:: $x$ : real, $k$ : 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

§32.3 Graphics

Contents

§32.3(i) First Painlevé Equation

§32.3(ii) Second Painlevé Equation with

§32.3(iii) Fourth Painlevé Equation with

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

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