§32.4 Isomonodromy Problems

§32.4(i) Definition

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

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

(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

$\mbox{P}_{\mbox{\scriptsize I}}$ 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

$\mbox{P}_{\mbox{\scriptsize II}}$ 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

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 $(\alpha,\beta,\gamma,\delta)=\left(2\theta_{0},2(1-\theta_{\infty}),1,-1\right),$

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

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).