Digital Library of Mathematical Functions
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32 Painlevé TranscendentsProperties

§32.3 Graphics

Contents

§32.3(i) First Painlevé Equation

Plots of solutions wk(x) of PI with wk(0)=0 and wk(0)=k for various values of k, and the parabola 6w2+x=0. For analytical explanation see §32.11(i).

See accompanying text
Figure 32.3.1: wk(x) for -12x1.33 and k=0.5, 0.75, 1, 1.25, and the parabola 6w2+x=0, shown in black. Magnify
See accompanying text
Figure 32.3.2: wk(x) for -12x2.43 and k=-0.5, -0.25, 0, 1, 2, and the parabola 6w2+x=0, shown in black. Magnify
See accompanying text
Figure 32.3.3: wk(x) for -12x0.73 and k=1.85185 3, 1.85185 5. The two graphs are indistinguishable when x exceeds -5.2, approximately. The parabola 6w2+x=0 is shown in black. Magnify
See accompanying text
Figure 32.3.4: wk(x) for -12x2.3 and k=-0.45142 7, -0.45142 8. The two graphs are indistinguishable when x exceeds -4.8, approximately. The parabola 6w2+x=0 is shown in black. Magnify

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

Here wk(x) is the solution of PII with α=0 and such that

32.3.1 wk(x)kAi(x),
x+;

compare §32.11(ii).

See accompanying text
Figure 32.3.5: wk(x) and kAi(x) for -10x4 with k=0.5. The two graphs are indistinguishable when x exceeds -0.4, approximately. Magnify
See accompanying text
Figure 32.3.6: wk(x) for -10x4 with k=0.999, 1.001. The two graphs are indistinguishable when x exceeds -2.8, approximately. The parabola 2w2+x=0 is shown in black. Magnify

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

Here u=uk(x;ν) is the solution of

32.3.2 2ux2=3u5+2xu3+(14x2-ν-12)u,

such that

32.3.3 ukU(-ν-12,x),
x+.

The corresponding solution of PIV is given by

32.3.4 w(x)=22uk2(2x,ν),

with β=0, α=2ν+1, and

32.3.5 w(x)22k2U2(-ν-12,2x),
x+;

compare (32.2.11) and §32.11(v). If we set 2u/x2=0 in (32.3.2) and solve for u, then

32.3.6 u2=-13x±16x2+12ν+6.
See accompanying text
Figure 32.3.7: uk(x;-12) for -12x4 with k=0.33554 691, 0.33554 692. The two graphs are indistinguishable when x exceeds -5.0, approximately. The parabolas u2+12x=0, u2+16x=0 are shown in black and green, respectively. Magnify
See accompanying text
Figure 32.3.8: uk(x;12) for -12x4 with k=0.47442, 0.47443. The two graphs are indistinguishable when x exceeds -2.2, approximately. The curves u2+13x±16x2+12=0 are shown in green and black, respectively. Magnify
See accompanying text
Figure 32.3.9: uk(x;32) for -12x4 with k=0.38736, 0.38737. The two graphs are indistinguishable when x exceeds -1.0, approximately. The curves u2+13x±16x2+24=0 are shown in green and black, respectively. Magnify
See accompanying text
Figure 32.3.10: uk(x;52) for -12x4 with k=0.24499 2, 0.24499 3. The two graphs are indistinguishable when x exceeds -0.6, approximately. The curves u2+13x±16x2+36=0 are shown in green and black, respectively. Magnify