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32 Painlevé TranscendentsProperties

§32.4 Isomonodromy Problems

Contents

§32.4(i) Definition

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Notes:
See Jimbo and Miwa (1981); also Fokas et al. (2006, Chapter 5), Its and Novokshënov (1986).
Keywords:
Lax pairs, Painlevé equations, Painlevé transcendents
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http://dlmf.nist.gov/32.4.i

\mbox{P}_{{\mbox{\scriptsize I}}}–\mbox{P}_{{\mbox{\scriptsize VI}}} can be expressed as the compatibility condition of a linear system, called an isomonodromy problem or Lax pair. Suppose

is a linear system in which \mathbf{A} and \mathbf{B} are matrices and \lambda is independent of z. Then the equation

32.4.2\frac{{\partial}^{2}\boldsymbol{{\Psi}}}{\partial z\partial\lambda}=\frac{{% \partial}^{2}\boldsymbol{{\Psi}}}{\partial\lambda\partial z},

is satisfied provided that

32.4.3\frac{\partial\mathbf{A}}{\partial z}-\frac{\partial\mathbf{B}}{\partial% \lambda}+\mathbf{A}\mathbf{B}-\mathbf{B}\mathbf{A}=0.

(32.4.3) is the compatibility condition of (32.4.1). Isomonodromy problems for Painlevé equations are not unique.

§32.4(ii) First Painlevé Equation

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Notes:
See Jimbo and Miwa (1981).
Permalink:
http://dlmf.nist.gov/32.4.ii

\mbox{P}_{{\mbox{\scriptsize I}}} is the compatibility condition of (32.4.1) with

32.4.4\mathbf{A}(z,\lambda)=(4\lambda^{4}+2w^{2}+z)\begin{bmatrix}1&0\\ 0&-1\end{bmatrix}-i(4\lambda^{2}w+2w^{2}+z)\begin{bmatrix}0&-i\\ i&0\end{bmatrix}-\left(2\lambda w^{{\prime}}+\frac{1}{2\lambda}\right)\begin{% bmatrix}0&1\\ 1&0\end{bmatrix},
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Symbols:
(a,b): open interval and z: real
Permalink:
http://dlmf.nist.gov/32.4.E4
Encodings:
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32.4.5\mathbf{B}(z,\lambda)=\left(\lambda+\dfrac{w}{\lambda}\right)\begin{bmatrix}1&% 0\\ 0&-1\end{bmatrix}-\dfrac{iw}{\lambda}\begin{bmatrix}0&-i\\ i&0\end{bmatrix}.
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Symbols:
(a,b): open interval and z: real
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http://dlmf.nist.gov/32.4.E5
Encodings:
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§32.4(iii) Second Painlevé Equation

\mbox{P}_{{\mbox{\scriptsize II}}} is the compatibility condition of (32.4.1) with

32.4.6\mathbf{A}(z,\lambda)=-i(4\lambda^{2}+2w^{2}+z)\begin{bmatrix}1&0\\ 0&-1\end{bmatrix}-2w^{{\prime}}\begin{bmatrix}0&-i\\ i&0\end{bmatrix}+\left(4\lambda w-\frac{\alpha}{\lambda}\right)\begin{bmatrix}% 0&1\\ 1&0\end{bmatrix},
32.4.7\mathbf{B}(z,\lambda)=\begin{bmatrix}-i\lambda&w\\ w&i\lambda\end{bmatrix}.
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Symbols:
(a,b): open interval and z: real
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http://dlmf.nist.gov/32.4.E7
Encodings:
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See Flaschka and Newell (1980).

§32.4(iv) Third Painlevé Equation

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Notes:
See Jimbo and Miwa (1981).
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http://dlmf.nist.gov/32.4.iv

The compatibility condition of (32.4.1) with

32.4.8\mathbf{A}(z,\lambda)=\begin{bmatrix}\tfrac{1}{4}z&0\\ 0&-\tfrac{1}{4}z\end{bmatrix}+\begin{bmatrix}-\tfrac{1}{2}\theta_{{\infty}}&u_% {0}\\ u_{1}&\tfrac{1}{2}\theta_{{\infty}}\end{bmatrix}\dfrac{1}{\lambda}+\begin{% bmatrix}v_{0}-\tfrac{1}{4}z&-v_{1}v_{0}\\ \ifrac{(v_{0}-\tfrac{1}{2}z)}{v_{1}}&\tfrac{1}{4}z-v_{0}\end{bmatrix}\frac{1}{% \lambda^{2}},
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Symbols:
(a,b): open interval, z: real and \theta_{\infty}: constant
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http://dlmf.nist.gov/32.4.E8
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32.4.9\mathbf{B}(z,\lambda)=\begin{bmatrix}\tfrac{1}{4}&0\\ 0&-\tfrac{1}{4}\end{bmatrix}\lambda+\begin{bmatrix}0&u_{0}\\ u_{1}&0\end{bmatrix}\dfrac{1}{z}-\begin{bmatrix}v_{0}-\tfrac{1}{4}z&-v_{1}v_{0% }\\ \ifrac{(v_{0}-\tfrac{1}{2}z)}{v_{1}}&\tfrac{1}{4}z-v_{0}\end{bmatrix}\frac{1}{% z\lambda},
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Symbols:
(a,b): open interval and z: real
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where \theta_{\infty} is an arbitrary constant, is

32.4.10zu_{0}^{{\prime}}=\theta_{{\infty}}u_{0}-zv_{0}v_{1},
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Symbols:
z: real and \theta_{\infty}: constant
Referenced by:
§32.4(iv)
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http://dlmf.nist.gov/32.4.E10
Encodings:
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32.4.11zu_{1}^{{\prime}}=-\theta_{{\infty}}u_{1}-(\ifrac{z(2v_{0}-z)}{(2v_{1})}),
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Symbols:
z: real and \theta_{\infty}: constant
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32.4.12zv_{0}^{{\prime}}=2v_{0}u_{1}v_{1}+v_{0}+(u_{0}(2v_{0}-z)/v_{1}),
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Symbols:
z: real
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32.4.13zv_{1}^{{\prime}}=2u_{0}-2u_{1}v_{1}^{2}-\theta_{{\infty}}v_{1}.
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Symbols:
z: real and \theta_{\infty}: constant
Referenced by:
§32.4(iv)
Permalink:
http://dlmf.nist.gov/32.4.E13
Encodings:
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If w=-u_{0}/(v_{0}v_{1}), then

32.4.14zw^{{\prime}}=(4v_{0}-z)w^{2}+(2\theta_{{\infty}}-1)w+z,
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Symbols:
z: real and \theta_{\infty}: constant
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http://dlmf.nist.gov/32.4.E14
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and w satisfies \mbox{P}_{{\mbox{\scriptsize III}}} with

where

32.4.16\theta_{0}=\frac{4v_{0}}{z}\left(\theta_{{\infty}}\left(1-\frac{z}{4v_{0}}% \right)+\frac{z-2v_{0}}{2v_{0}v_{1}}u_{0}+u_{1}v_{1}\right).
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Symbols:
z: real, \theta_{\infty}: constant and \theta_{0}
Permalink:
http://dlmf.nist.gov/32.4.E16
Encodings:
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Note that the right-hand side of the last equation is a first integral of the system (32.4.10)–(32.4.13).

§32.4(v) Other Painlevé Equations

For isomonodromy problems for \mbox{P}_{{\mbox{\scriptsize IV}}}, \mbox{P}_{{\mbox{\scriptsize V}}}, and \mbox{P}_{{\mbox{\scriptsize VI}}} see Jimbo and Miwa (1981).

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