32 Painlevé Transcendents Properties 32.2 Differential Equations 32.4 Isomonodromy Problems

§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 , 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

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

Symbols:: : real and : 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

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

Symbols:: : real and : 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

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

Symbols:: : real and : 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

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

Symbols:: : real and : 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, : real and : 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

. The two graphs are indistinguishable when

exceeds −0.4, approximately.

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

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.

Symbols:: : real and : 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 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

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, : real, : 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:: : real, : 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, : real, : 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 , then

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

Symbols:: : 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

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:: : 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

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.

Symbols:: : real, : 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

, 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.

Symbols:: : real, : 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

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:: : 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

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 32.2 Differential Equations 32.4 Isomonodromy Problems

§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 $\beta=0$