§32.4 Isomonodromy Problems

Permalink:: http://dlmf.nist.gov/32.4

§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

Notes:: See Jimbo and Miwa (1981); also Fokas et al. (2006, Chapter 5), Its and Novokshënov (1986).
Keywords:: Lax pairs, Painlevé equations, Painlevé transcendents
Permalink:: 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

32.4.1

$\frac{\partial\boldsymbol{{\Psi}}}{\partial\lambda}=\mathbf{A}(z,\lambda)\boldsymbol{{\Psi}},$

$\frac{\partial\boldsymbol{{\Psi}}}{\partial z}=\mathbf{B}(z,\lambda)\boldsymbol{{\Psi}},$

Symbols:: $\frac{\partial f}{\partial x}$ : partial derivative of $f$ with respect to $x$ 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

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:: $\partial x$ : partial differential of $x$ and $z$ : real
Permalink:: http://dlmf.nist.gov/32.4.E2
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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:: $\frac{\partial f}{\partial x}$ : partial derivative of $f$ with respect to $x$ and $z$ : real
Referenced by:: §32.4(i)
Permalink:: http://dlmf.nist.gov/32.4.E3
Encodings:: TeX, pMML, png

(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

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 $\mathbf{A}(z,\lambda)=(4\lambda^{4}+2w^{2}+z)\begin{bmatrix}1&0\\ 0&-1\end{bmatrix}-i(4\lambda^{2}w+2w^{2}+z)\begin{bmatrix}0&-i\\ i&0\end{bmatrix}-\left(2\lambda w^{{\prime}}+\frac{1}{2\lambda}\right)\begin{bmatrix}0&1\\ 1&0\end{bmatrix},$

Symbols:: $(a,b)$ : open interval and $z$ : real
Permalink:: http://dlmf.nist.gov/32.4.E4
Encodings:: TeX, pMML, png

32.4.5 $\mathbf{B}(z,\lambda)=\left(\lambda+\dfrac{w}{\lambda}\right)\begin{bmatrix}1&0\\ 0&-1\end{bmatrix}-\dfrac{iw}{\lambda}\begin{bmatrix}0&-i\\ i&0\end{bmatrix}.$

Symbols:: $(a,b)$ : open interval and $z$ : real
Permalink:: http://dlmf.nist.gov/32.4.E5
Encodings:: TeX, pMML, png

§32.4(iii) Second Painlevé Equation

Permalink:: http://dlmf.nist.gov/32.4.iii

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

32.4.6 $\mathbf{A}(z,\lambda)=-i(4\lambda^{2}+2w^{2}+z)\begin{bmatrix}1&0\\ 0&-1\end{bmatrix}-2w^{{\prime}}\begin{bmatrix}0&-i\\ i&0\end{bmatrix}+\left(4\lambda w-\frac{\alpha}{\lambda}\right)\begin{bmatrix}0&1\\ 1&0\end{bmatrix},$

Symbols:: $(a,b)$ : open interval, $z$ : real and $\alpha$ : arbitrary constant
Permalink:: http://dlmf.nist.gov/32.4.E6
Encodings:: TeX, pMML, png

32.4.7 $\mathbf{B}(z,\lambda)=\begin{bmatrix}-i\lambda&w\\ w&i\lambda\end{bmatrix}.$

Symbols:: $(a,b)$ : open interval and $z$ : real
Permalink:: http://dlmf.nist.gov/32.4.E7
Encodings:: TeX, pMML, png

See Flaschka and Newell (1980).

§32.4(iv) Third Painlevé Equation

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

The compatibility condition of (32.4.1) with

32.4.8 $\mathbf{A}(z,\lambda)=\begin{bmatrix}\tfrac{1}{4}z&0\\ 0&-\tfrac{1}{4}z\end{bmatrix}+\begin{bmatrix}-\tfrac{1}{2}\theta _{{\infty}}&u_{0}\\ u_{1}&\tfrac{1}{2}\theta _{{\infty}}\end{bmatrix}\dfrac{1}{\lambda}+\begin{bmatrix}v_{0}-\tfrac{1}{4}z&-v_{1}v_{0}\\ \ifrac{(v_{0}-\tfrac{1}{2}z)}{v_{1}}&\tfrac{1}{4}z-v_{0}\end{bmatrix}\frac{1}{\lambda^{2}},$

Symbols:: $(a,b)$ : open interval, $z$ : real and $\theta _{\infty}$ : constant
Permalink:: http://dlmf.nist.gov/32.4.E8
Encodings:: TeX, pMML, png

32.4.9 $\mathbf{B}(z,\lambda)=\begin{bmatrix}\tfrac{1}{4}&0\\ 0&-\tfrac{1}{4}\end{bmatrix}\lambda+\begin{bmatrix}0&u_{0}\\ u_{1}&0\end{bmatrix}\dfrac{1}{z}-\begin{bmatrix}v_{0}-\tfrac{1}{4}z&-v_{1}v_{0}\\ \ifrac{(v_{0}-\tfrac{1}{2}z)}{v_{1}}&\tfrac{1}{4}z-v_{0}\end{bmatrix}\frac{1}{z\lambda},$

Symbols:: $(a,b)$ : open interval and $z$ : real
Permalink:: http://dlmf.nist.gov/32.4.E9
Encodings:: TeX, pMML, png

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

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

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

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

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
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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
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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

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

Keywords:: Painlevé equations
Permalink:: http://dlmf.nist.gov/32.4.v

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).