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32 Painlevé TranscendentsProperties

§32.6 Hamiltonian Structure

Contents

§32.6(i) Introduction

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Notes:
See Okamoto (1981).
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http://dlmf.nist.gov/32.6.i

\mbox{P}_{{\mbox{\scriptsize I}}}–\mbox{P}_{{\mbox{\scriptsize VI}}} can be written as a Hamiltonian system

for suitable (non-autonomous) Hamiltonian functions \mathrm{H}(q,p,z).

§32.6(ii) First Painlevé Equation

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Notes:
See Jimbo and Miwa (1981), Okamoto (1981).
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http://dlmf.nist.gov/32.6.ii

The Hamiltonian for \mbox{P}_{{\mbox{\scriptsize I}}} is

32.6.2\mathrm{H}_{{\mbox{\scriptsize I}}}(q,p,z)=\tfrac{1}{2}p^{2}-2q^{3}-zq,
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Symbols:
z: real, q: coordinate, p: momentum and \mathrm{H}(q,p,z): Hamiltonian
Referenced by:
§32.6(ii)
Permalink:
http://dlmf.nist.gov/32.6.E2
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and so

32.6.3q^{{\prime}}=p,
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Symbols:
q: coordinate and p: momentum
Referenced by:
§32.6(ii)
Permalink:
http://dlmf.nist.gov/32.6.E3
Encodings:
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32.6.4p^{{\prime}}=6q^{2}+z.
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Symbols:
z: real, q: coordinate and p: momentum
Referenced by:
§32.6(ii)
Permalink:
http://dlmf.nist.gov/32.6.E4
Encodings:
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Then q=w satisfies \mbox{P}_{{\mbox{\scriptsize I}}}. The function

defined by (32.6.2) satisfies

32.6.6\left(\sigma^{{\prime\prime}}\right)^{2}+4\left(\sigma^{{\prime}}\right)^{3}+2% z\sigma^{{\prime}}-2\sigma=0.
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Symbols:
z: real and \sigma: function
Referenced by:
§32.6(ii)
Permalink:
http://dlmf.nist.gov/32.6.E6
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Conversely, if \sigma is a solution of (32.6.6), then

32.6.7q=-\sigma^{{\prime}},
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Symbols:
q: coordinate and \sigma: function
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http://dlmf.nist.gov/32.6.E7
Encodings:
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32.6.8p=-\sigma^{{\prime\prime}},
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Symbols:
p: momentum and \sigma: function
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http://dlmf.nist.gov/32.6.E8
Encodings:
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are solutions of (32.6.3) and (32.6.4).

§32.6(iii) Second Painlevé Equation

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Notes:
See Jimbo and Miwa (1981), Okamoto (1986).
Permalink:
http://dlmf.nist.gov/32.6.iii

The Hamiltonian for \mbox{P}_{{\mbox{\scriptsize II}}} is

and so

32.6.10q^{{\prime}}=p-q^{2}-\tfrac{1}{2}z,
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Symbols:
z: real, q: coordinate and p: momentum
Referenced by:
§32.6(iii)
Permalink:
http://dlmf.nist.gov/32.6.E10
Encodings:
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32.6.11p^{{\prime}}=2qp+\alpha+\tfrac{1}{2}.
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Symbols:
\alpha: arbitrary constant, q: coordinate and p: momentum
Referenced by:
§32.6(iii)
Permalink:
http://dlmf.nist.gov/32.6.E11
Encodings:
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Then q=w satisfies \mbox{P}_{{\mbox{\scriptsize II}}} and p satisfies

32.6.12pp^{{\prime\prime}}=\tfrac{1}{2}(p^{{\prime}})^{2}+2p^{3}-zp^{2}-\tfrac{1}{2}(% \alpha+\tfrac{1}{2})^{2}.

The function \sigma(z)=\mathrm{H}_{{\mbox{\scriptsize II}}}(q,p,z) defined by (32.6.9) satisfies

32.6.13\left(\sigma^{{\prime\prime}}\right)^{2}+4\left(\sigma^{{\prime}}\right)^{3}+2% \sigma^{{\prime}}\left(z\sigma^{{\prime}}-\sigma\right)=\tfrac{1}{4}(\alpha+% \tfrac{1}{2})^{2}.
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Symbols:
z: real, \alpha: arbitrary constant and \sigma(z): function
Referenced by:
§32.6(iii)
Permalink:
http://dlmf.nist.gov/32.6.E13
Encodings:
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Conversely, if \sigma(z) is a solution of (32.6.13), then

32.6.14q=\ifrac{(4\sigma^{{\prime\prime}}+2\alpha+1)}{(8\sigma^{{\prime}})},
32.6.15p=-2\sigma^{{\prime}},
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Symbols:
p: momentum and \sigma(z): function
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http://dlmf.nist.gov/32.6.E15
Encodings:
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are solutions of (32.6.10) and (32.6.11).

§32.6(iv) Third Painlevé Equation

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Notes:
See Jimbo and Miwa (1981), Okamoto (1987c); also Forrester and Witte (2002).
Permalink:
http://dlmf.nist.gov/32.6.iv

The Hamiltonian for \mbox{P}_{{\mbox{\scriptsize III}}} is

32.6.16z\mathrm{H}_{{\mbox{\scriptsize III}}}(q,p,z)=q^{2}p^{2}-{\left(\kappa_{{% \infty}}zq^{2}+(2\theta_{0}+1)q-\kappa_{0}z\right)p}+\kappa_{{\infty}}(\theta_% {0}+\theta_{{\infty}})zq,

and so

32.6.17zq^{{\prime}}=2q^{2}p-\kappa_{{\infty}}zq^{2}-(2\theta_{0}+1)q+\kappa_{0}z,
32.6.18zp^{{\prime}}=-2qp^{2}+2\kappa_{{\infty}}zqp+(2\theta_{0}+1)p-\kappa_{{\infty}% }(\theta_{0}+\theta_{\infty})z.

Then q=w satisfies \mbox{P}_{{\mbox{\scriptsize III}}} with

The function

defined by (32.6.16) satisfies

32.6.21(z\sigma^{{\prime\prime}}-\sigma^{{\prime}})^{2}+2\left((\sigma^{{\prime}})^{2% }-\kappa_{0}^{2}\kappa_{{\infty}}^{2}z^{2}\right)(z\sigma^{{\prime}}-2\sigma)+% 8\kappa_{0}\kappa_{{\infty}}\theta_{0}\theta_{{\infty}}z\sigma^{{\prime}}=4% \kappa_{0}^{2}\kappa_{{\infty}}^{2}(\theta_{0}^{2}+\theta_{{\infty}}^{2})z^{2}.
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Symbols:
z: real, \sigma: function, \theta: parameter and \kappa_{j}: parameters
Referenced by:
§32.6(iv)
Permalink:
http://dlmf.nist.gov/32.6.E21
Encodings:
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Conversely, if \sigma is a solution of (32.6.21), then

32.6.23p=\ifrac{(\sigma^{{\prime}}+\kappa_{0}\kappa_{{\infty}}z)}{(2\kappa_{0})},

are solutions of (32.6.17) and (32.6.18).

The Hamiltonian for \mbox{P}^{{\prime}}_{{\mbox{\scriptsize III}}} (§32.2(iii)) is

32.6.24\zeta\mathrm{H}_{{\mbox{\scriptsize III}}}(q,p,\zeta)=q^{2}p^{2}-\left(\eta_{{% \infty}}q^{2}+\theta_{0}q-\eta_{0}\zeta\right)p+\tfrac{1}{2}\eta_{{\infty}}(% \theta_{0}+\theta_{{\infty}})q,
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Symbols:
q: coordinate, \mathrm{H}(q,p,z): Hamiltonian, \theta: parameter and p: momentum
Referenced by:
§32.6(iv)
Permalink:
http://dlmf.nist.gov/32.6.E24
Encodings:
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and so

32.6.25\zeta q^{{\prime}}=2q^{2}p-\eta_{{\infty}}q^{2}-\theta_{0}q+\eta_{0}\zeta,
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Symbols:
q: coordinate, \theta: parameter and p: momentum
Referenced by:
§32.6(iv)
Permalink:
http://dlmf.nist.gov/32.6.E25
Encodings:
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32.6.26\zeta p^{{\prime}}=-2qp^{2}+2\eta_{{\infty}}qp+\theta_{0}p-\tfrac{1}{2}\eta_{{% \infty}}(\theta_{0}+\theta_{1}).
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Symbols:
q: coordinate, \theta: parameter and p: momentum
Referenced by:
§32.6(iv)
Permalink:
http://dlmf.nist.gov/32.6.E26
Encodings:
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Then q=u satisfies \mbox{P}^{{\prime}}_{{\mbox{\scriptsize III}}} with

The function

defined by (32.6.24) satisfies

32.6.29\zeta^{2}(\sigma^{{\prime\prime}})^{2}+\left(4(\sigma^{{\prime}})^{2}-\eta_{0}% ^{2}\eta_{{\infty}}^{2}\right)(\zeta\sigma^{{\prime}}-\sigma)+\eta_{0}\eta_{{% \infty}}\theta_{0}\theta_{{\infty}}\sigma^{{\prime}}=\tfrac{1}{4}\eta_{0}^{2}% \eta_{{\infty}}^{2}(\theta_{0}^{2}+\theta_{{\infty}}^{2}).
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Symbols:
\sigma: function and \theta: parameter
Referenced by:
§32.6(iv)
Permalink:
http://dlmf.nist.gov/32.6.E29
Encodings:
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Conversely, if \sigma is a solution of (32.6.29), then

32.6.30q=\frac{\eta_{0}\left(\zeta\sigma^{{\prime\prime}}-2\theta_{0}\sigma^{{\prime}% }+\eta_{0}\eta_{{\infty}}\theta_{{\infty}}\right)}{\eta_{0}^{2}\eta_{{\infty}}% ^{2}-4(\sigma^{{\prime}})^{2}},
32.6.31p=\ifrac{(2\sigma^{{\prime}}+\eta_{0}\eta_{{\infty}}\zeta)}{(2\eta_{0})},
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Symbols:
\sigma: function and p: momentum
Permalink:
http://dlmf.nist.gov/32.6.E31
Encodings:
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are solutions of (32.6.25) and (32.6.26).

The Hamiltonian for \mbox{P}_{{\mbox{\scriptsize III}}} with \gamma=0 is

32.6.32z\mathrm{H}_{{\mbox{\scriptsize III}}}(q,p,z)=q^{2}p^{2}+(\theta q-\kappa_{0}z% )p-\kappa_{{\infty}}zq,

and so

32.6.33zq^{{\prime}}=2q^{2}p+\theta q-\kappa_{0}z,
32.6.34zp^{{\prime}}=-2qp^{2}-\theta p+\kappa_{{\infty}}z.

Then q=w satisfies \mbox{P}_{{\mbox{\scriptsize III}}} with

The function

defined by (32.6.32) satisfies

32.6.37(z\sigma^{{\prime\prime}}-\sigma^{{\prime}})^{2}+2(\sigma^{{\prime}})^{2}(z% \sigma^{{\prime}}-2\sigma)-4\kappa_{0}\kappa_{{\infty}}(\theta+1)\theta_{{% \infty}}z\sigma^{{\prime}}=4\kappa_{0}^{2}\kappa_{{\infty}}^{2}z^{2}.
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Symbols:
z: real, \sigma: function, \theta: parameter and \kappa_{j}: parameters
Referenced by:
§32.6(iv)
Permalink:
http://dlmf.nist.gov/32.6.E37
Encodings:
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Conversely, if \sigma is a solution of (32.6.37), then

32.6.39p=\ifrac{\sigma^{{\prime}}}{(2\kappa_{0})},

are solutions of (32.6.33) and (32.6.34).

§32.6(v) Other Painlevé Equations

For Hamiltonian structure for \mbox{P}_{{\mbox{\scriptsize IV}}} see Jimbo and Miwa (1981), Okamoto (1986); also Forrester and Witte (2001).

For Hamiltonian structure for \mbox{P}_{{\mbox{\scriptsize V}}} see Jimbo and Miwa (1981), Okamoto (1987b); also Forrester and Witte (2002).

For Hamiltonian structure for \mbox{P}_{{\mbox{\scriptsize VI}}} see Jimbo and Miwa (1981) and Okamoto (1987a); also Forrester and Witte (2004).

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