32 Painlevé Transcendents Properties 32.3 Graphics 32.5 Integral Equations

§32.4 Isomonodromy Problems

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Permalink:: http://dlmf.nist.gov/32.4
See also:: Annotations for Ch.32

§32.4(i) Definition
§32.4(ii) First Painlevé Equation
§32.4(iii) Second Painlevé Equation
§32.4(iv) Third Painlevé Equation
§32.4(v) Other Painlevé Equations

§32.4(i) Definition

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Keywords:: Lax pair, Lax pairs, Painlevé equations, Painlevé transcendents, compatibility condition, compatibility conditions, isomonodromy problems
Notes:: See Jimbo and Miwa (1981); also Fokas et al. (2006, Chapter 5), Its and Novokshënov (1986).
Permalink:: http://dlmf.nist.gov/32.4.i
See also:: Annotations for §32.4 and Ch.32

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

32.4.1	$\displaystyle\frac{\partial\boldsymbol{{\Psi}}}{\partial\lambda}$	$\displaystyle=\mathbf{A}(z,\lambda)\boldsymbol{{\Psi}},$
32.4.1	$\displaystyle\frac{\partial\boldsymbol{{\Psi}}}{\partial z}$	$\displaystyle=\mathbf{B}(z,\lambda)\boldsymbol{{\Psi}},$
ⓘ Symbols: $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ and $z$ : real Referenced by: §32.4(i), §32.4(ii), §32.4(iii), §32.4(iv) Permalink: http://dlmf.nist.gov/32.4.E1 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §32.4(i), §32.4 and Ch.32

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		$\frac{\,{\partial}^{2}\boldsymbol{{\Psi}}}{\,\partial z\,\partial\lambda}=% \frac{\,{\partial}^{2}\boldsymbol{{\Psi}}}{\,\partial\lambda\,\partial z},$
ⓘ Symbols: $\,\partial\NVar{x}$ : partial differential of $x$ and $z$ : real Permalink: http://dlmf.nist.gov/32.4.E2 Encodings: TeX, pMML, png See also: Annotations for §32.4(i), §32.4 and Ch.32

is satisfied provided that

32.4.3		$\frac{\partial\mathbf{A}}{\partial z}-\frac{\partial\mathbf{B}}{\partial% \lambda}+\mathbf{A}\mathbf{B}-\mathbf{B}\mathbf{A}=0.$
ⓘ Symbols: $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ and $z$ : real Referenced by: §32.4(i) Permalink: http://dlmf.nist.gov/32.4.E3 Encodings: TeX, pMML, png See also: Annotations for §32.4(i), §32.4 and Ch.32

(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
See also:: Annotations for §32.4 and Ch.32

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

32.4.4		$\mathbf{A}(z,\lambda)=(4\lambda^{4}+2w^{2}+z)\begin{bmatrix}1&0\\ 0&-1\end{bmatrix}-i(4\lambda^{2}w+2w^{2}+z)\begin{bmatrix}0&-i\\ i&0\end{bmatrix}-\left(2\lambda w^{\prime}+\frac{1}{2\lambda}\right)\begin{% bmatrix}0&1\\ 1&0\end{bmatrix},$
ⓘ Defines: $\mathbf{A}(\NVar{z},\NVar{\lambda})$ : matrix (locally) Symbols: $\mathrm{i}$ : imaginary unit and $z$ : real Permalink: http://dlmf.nist.gov/32.4.E4 Encodings: TeX, pMML, png See also: Annotations for §32.4(ii), §32.4 and Ch.32

32.4.5		$\mathbf{B}(z,\lambda)=\left(\lambda+\dfrac{w}{\lambda}\right)\begin{bmatrix}1&% 0\\ 0&-1\end{bmatrix}-\dfrac{iw}{\lambda}\begin{bmatrix}0&-i\\ i&0\end{bmatrix}.$
ⓘ Defines: $\mathbf{B}(\NVar{z},\NVar{\lambda})$ : matrix (locally) Symbols: $\mathrm{i}$ : imaginary unit and $z$ : real Permalink: http://dlmf.nist.gov/32.4.E5 Encodings: TeX, pMML, png See also: Annotations for §32.4(ii), §32.4 and Ch.32

§32.4(iii) Second Painlevé Equation

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Permalink:: http://dlmf.nist.gov/32.4.iii
See also:: Annotations for §32.4 and Ch.32

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

32.4.6		$\mathbf{A}(z,\lambda)=-i(4\lambda^{2}+2w^{2}+z)\begin{bmatrix}1&0\\ 0&-1\end{bmatrix}-2w^{\prime}\begin{bmatrix}0&-i\\ i&0\end{bmatrix}+\left(4\lambda w-\frac{\alpha}{\lambda}\right)\begin{bmatrix}% 0&1\\ 1&0\end{bmatrix},$
ⓘ Defines: $\mathbf{A}(\NVar{z},\NVar{\lambda})$ : matrix (locally) Symbols: $\mathrm{i}$ : imaginary unit, $z$ : real and $\alpha$ : arbitrary constant Permalink: http://dlmf.nist.gov/32.4.E6 Encodings: TeX, pMML, png See also: Annotations for §32.4(iii), §32.4 and Ch.32

32.4.7		$\mathbf{B}(z,\lambda)=\begin{bmatrix}-i\lambda&w\\ w&i\lambda\end{bmatrix}.$
ⓘ Defines: $\mathbf{B}(\NVar{z},\NVar{\lambda})$ : matrix (locally) Symbols: $\mathrm{i}$ : imaginary unit and $z$ : real Permalink: http://dlmf.nist.gov/32.4.E7 Encodings: TeX, pMML, png See also: Annotations for §32.4(iii), §32.4 and Ch.32

See Flaschka and Newell (1980).

§32.4(iv) Third Painlevé Equation

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Notes:: See Jimbo and Miwa (1981).
Permalink:: http://dlmf.nist.gov/32.4.iv
See also:: Annotations for §32.4 and Ch.32

The compatibility condition of (32.4.1) with

32.4.8		$\mathbf{A}(z,\lambda)=\begin{bmatrix}\tfrac{1}{4}z&0\\ 0&-\tfrac{1}{4}z\end{bmatrix}+\begin{bmatrix}-\tfrac{1}{2}\theta_{\infty}&u_{0% }\\ u_{1}&\tfrac{1}{2}\theta_{\infty}\end{bmatrix}\dfrac{1}{\lambda}+\begin{% bmatrix}v_{0}-\tfrac{1}{4}z&-v_{1}v_{0}\\ \ifrac{(v_{0}-\tfrac{1}{2}z)}{v_{1}}&\tfrac{1}{4}z-v_{0}\end{bmatrix}\frac{1}{% \lambda^{2}},$
ⓘ Defines: $\mathbf{A}(\NVar{z},\NVar{\lambda})$ : matrix (locally) Symbols: $z$ : real and $\theta_{\infty}$ : constant Permalink: http://dlmf.nist.gov/32.4.E8 Encodings: TeX, pMML, png See also: Annotations for §32.4(iv), §32.4 and Ch.32

32.4.9		$\mathbf{B}(z,\lambda)=\begin{bmatrix}\tfrac{1}{4}&0\\ 0&-\tfrac{1}{4}\end{bmatrix}\lambda+\begin{bmatrix}0&u_{0}\\ u_{1}&0\end{bmatrix}\dfrac{1}{z}-\begin{bmatrix}v_{0}-\tfrac{1}{4}z&-v_{1}v_{0% }\\ \ifrac{(v_{0}-\tfrac{1}{2}z)}{v_{1}}&\tfrac{1}{4}z-v_{0}\end{bmatrix}\frac{1}{% z\lambda},$
ⓘ Defines: $\mathbf{B}(\NVar{z},\NVar{\lambda})$ : matrix (locally) Symbols: $z$ : real Permalink: http://dlmf.nist.gov/32.4.E9 Encodings: TeX, pMML, png See also: Annotations for §32.4(iv), §32.4 and Ch.32

where $\theta_{\infty}$ is an arbitrary constant, is

32.4.10		$zu_{0}^{\prime}=\theta_{\infty}u_{0}-zv_{0}v_{1},$
ⓘ Symbols: $z$ : real and $\theta_{\infty}$ : constant Referenced by: §32.4(iv) Permalink: http://dlmf.nist.gov/32.4.E10 Encodings: TeX, pMML, png See also: Annotations for §32.4(iv), §32.4 and Ch.32

32.4.11		$zu_{1}^{\prime}=-\theta_{\infty}u_{1}-(\ifrac{z(2v_{0}-z)}{(2v_{1})}),$
ⓘ Symbols: $z$ : real and $\theta_{\infty}$ : constant Permalink: http://dlmf.nist.gov/32.4.E11 Encodings: TeX, pMML, png See also: Annotations for §32.4(iv), §32.4 and Ch.32

32.4.12		$zv_{0}^{\prime}=2v_{0}u_{1}v_{1}+v_{0}+(u_{0}(2v_{0}-z)/v_{1}),$
ⓘ Symbols: $z$ : real Permalink: http://dlmf.nist.gov/32.4.E12 Encodings: TeX, pMML, png See also: Annotations for §32.4(iv), §32.4 and Ch.32

32.4.13		$zv_{1}^{\prime}=2u_{0}-2u_{1}v_{1}^{2}-\theta_{\infty}v_{1}.$
ⓘ Symbols: $z$ : real and $\theta_{\infty}$ : constant Referenced by: §32.4(iv) Permalink: http://dlmf.nist.gov/32.4.E13 Encodings: TeX, pMML, png See also: Annotations for §32.4(iv), §32.4 and Ch.32

If $w=-u_{0}/(v_{0}v_{1})$ , then

32.4.14		$zw^{\prime}=(4v_{0}-z)w^{2}+(2\theta_{\infty}-1)w+z,$
ⓘ Symbols: $z$ : real and $\theta_{\infty}$ : constant Permalink: http://dlmf.nist.gov/32.4.E14 Encodings: TeX, pMML, png See also: Annotations for §32.4(iv), §32.4 and Ch.32

and $w$ satisfies $\mbox{P}_{\mbox{\scriptsize III}}$ with

32.4.15		$(\alpha,\beta,\gamma,\delta)=\left(2\theta_{0},2(1-\theta_{\infty}),1,-1\right),$
ⓘ Symbols: $\alpha$ : arbitrary constant, $\beta$ : arbitrary constant, $\gamma$ : arbitrary constant, $\delta$ : arbitrary constant, $\theta_{\infty}$ : constant and $\theta_{0}$ Permalink: http://dlmf.nist.gov/32.4.E15 Encodings: TeX, pMML, png See also: Annotations for §32.4(iv), §32.4 and Ch.32

where

32.4.16		$\theta_{0}=\frac{4v_{0}}{z}\left(\theta_{\infty}\left(1-\frac{z}{4v_{0}}\right% )+\frac{z-2v_{0}}{2v_{0}v_{1}}u_{0}+u_{1}v_{1}\right).$
ⓘ Symbols: $z$ : real, $\theta_{\infty}$ : constant and $\theta_{0}$ Permalink: http://dlmf.nist.gov/32.4.E16 Encodings: TeX, pMML, png See also: Annotations for §32.4(iv), §32.4 and Ch.32

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

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Keywords:: Painlevé equations, compatibility conditions
Permalink:: http://dlmf.nist.gov/32.4.v
See also:: Annotations for §32.4 and Ch.32

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).

32.3 Graphics 32.5 Integral Equations

NIST

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