§32.4 Isomonodromy Problems

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

32.4.1		${\displaystyle \frac{\partial \mathbf{\Psi }}{\partial \lambda }}$	$=\mathbf{A}(z,\lambda )\mathbf{\Psi },$
		${\displaystyle \frac{\partial \mathbf{\Psi }}{\partial z}}$	$=\mathbf{B}(z,\lambda )\mathbf{\Psi },$
ⓘ Symbols: $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$, $\partial x$: 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		$$\frac{{\partial }^2\mathbf{\Psi }}{\partial z\partial \lambda }=\frac{{\partial }^2\mathbf{\Psi }}{\partial \lambda \partial z},$$
ⓘ Symbols: $\partial x$: 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: $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$, $\partial x$: 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.

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

32.4.4
ⓘ Symbols: $\mathrm{i}$: imaginary unit, $(a,b)$: open interval 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
ⓘ Symbols: $\mathrm{i}$: imaginary unit, $(a,b)$: open interval 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

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

32.4.6
ⓘ Symbols: $\mathrm{i}$: imaginary unit, $(a,b)$: open interval, $z$: 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
ⓘ Symbols: $\mathrm{i}$: imaginary unit, $(a,b)$: open interval 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).

The compatibility condition of (32.4.1) with

32.4.8
ⓘ Symbols: $(a,b)$: open interval, $z$: real and ${\theta }_{\mathrm{\infty }}$: 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
ⓘ Symbols: $(a,b)$: open interval and $z$: 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 ${\theta }_{\mathrm{\infty }}$ is an arbitrary constant, is

32.4.10		$$z{u}_0^{\prime }={\theta }_{\mathrm{\infty }}{u}_0-z{v}_0{v}_1,$$
ⓘ Symbols: $z$: real and ${\theta }_{\mathrm{\infty }}$: 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		$$z{u}_1^{\prime }=-{\theta }_{\mathrm{\infty }}{u}_1-(z(2v_0-z)/(2v_1)),$$
ⓘ Symbols: $z$: real and ${\theta }_{\mathrm{\infty }}$: 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		$$z{v}_0^{\prime }=2v_0{u}_1{v}_1+v_0+(u_0(2v_0-z)/v_1),$$
ⓘ 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		$$z{v}_1^{\prime }=2u_0-2u_1{v}_1^2-{\theta }_{\mathrm{\infty }}{v}_1.$$
ⓘ Symbols: $z$: real and ${\theta }_{\mathrm{\infty }}$: 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		$$z{w}^{\prime }=(4v_0-z)w^2+(2{\theta }_{\mathrm{\infty }}-1)w+z,$$
ⓘ Symbols: $z$: real and ${\theta }_{\mathrm{\infty }}$: 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 ${\text{P}}_{\text{III}}$ with

32.4.15		$$(\alpha ,\beta ,\gamma ,\delta )=(2{\theta }_0,2(1-{\theta }_{\mathrm{\infty }}),1,-1),$$
ⓘ Symbols: $\alpha$: arbitrary constant, $\beta$: arbitrary constant, $\gamma$: arbitrary constant, $\delta$: arbitrary constant, ${\theta }_{\mathrm{\infty }}$: 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		$${\theta }_0=\frac{4v_0}{z}\left({\theta }_{\mathrm{\infty }}\left(1-\frac{z}{4v_0}\right)+\frac{z-2v_0}{2v_0{v}_1}{u}_0+u_1{v}_1\right).$$
ⓘ Symbols: $z$: real, ${\theta }_{\mathrm{\infty }}$: 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).

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