New $2\times 2$-Matrix Linear Problems for the Painlevé Equations III, V

Abstract

We construct $2\times 2$-matrix linear problems with a spectral parameter for the Painlevé equations I–V by means of the degeneration processes from the elliptic linear problem for the Painlevé VI equation. These processes supplement the known degeneration relations between the Painlevé equations with the degeneration scheme for the associated linear problems. The degeneration relations constructed in this paper are based on the trigonometric, rational, and Inozemtsev limits. The obtained $2\times 2$-matrix linear problems for the Painlevé equations III and V are new.

Appendices

Appendix 1: Painlevé Equations

In this section, we present a connection between the different forms of the Painlevé equations V, IV, and III considered in this paper.

1.1 Painlevé V

The Painlevé V equation has the following rational form:

$$\begin{aligned} \dfrac{\mathrm{d}^2\lambda }{\mathrm{d}t^2}= & {} \left( \dfrac{1}{2\lambda }+\dfrac{1}{\lambda -1}\right) \left( \dfrac{\mathrm{d}\lambda }{\mathrm{d}t}\right) ^2-\dfrac{1}{t}\dfrac{\mathrm{d}\lambda }{\mathrm{d}t}+\dfrac{\left( \lambda -1\right) ^2}{t^2}\left( \alpha \lambda +\dfrac{\beta }{\lambda }\right) +\gamma \dfrac{\lambda }{t}\nonumber \\&+\,\delta \dfrac{\lambda \left( \lambda +1\right) }{\left( \lambda -1\right) }. \end{aligned}$$

(8.1)

In order to obtain the equivalent form of (8.1) which we use in Sect. 2, one can perform the following change of variables:

$$\begin{aligned} \lambda \left( t\right) =\lambda \left( u\left( t\right) \right) =-\tan ^2\left( \pi u\left( t\right) \right) . \end{aligned}$$

As a result, $\left( \mathrm{d}\lambda /\mathrm{d}t\right) ^2$ becomes zero. After the substitution $t\left( \tau \right) =\mathrm{e}^{\tau }$, the derivative $\mathrm{d}\lambda /\mathrm{d}t$ becomes zero as well, which leads to

$$\begin{aligned} \dfrac{\mathrm{d}^2 u}{\mathrm{d}t^2}=\dfrac{\alpha }{2\pi }\dfrac{\sin \left( \pi u\right) }{\cos ^3\left( \pi u\right) }+\dfrac{\beta }{2\pi }\dfrac{\cos \left( \pi u\right) }{\sin ^3\left( \pi u\right) }+2\gamma \mathrm{e}^{\tau }\sin \left( 2\pi u\right) +\delta \mathrm{e}^{2\tau }\sin \left( 4\pi u\right) .\nonumber \\ \end{aligned}$$

(8.2)

1.2 Painlevé IV

The Painlevé IV equation has the following rational form:

$$\begin{aligned} \dfrac{\mathrm{d}^2\lambda }{\mathrm{d}t^2}=\dfrac{1}{2\lambda }\left( \dfrac{\mathrm{d}\lambda }{\mathrm{d}t}\right) ^2+\dfrac{3}{2}\lambda ^3+4t\lambda ^2+2\left( t^2-\alpha \right) \lambda +\dfrac{\beta }{\lambda }. \end{aligned}$$

(8.3)

In order to obtain the equivalent form of (8.3) considered in Sect. 3, one can make the change of variables

$$\begin{aligned} \lambda \left( t\right) =u^2\left( t\right) , \end{aligned}$$

which leads to

$$\begin{aligned} \dfrac{\mathrm{d}^2 u}{\mathrm{d}t^2}=\dfrac{3u^5}{4}+2tu^3+\left( t^2-\alpha \right) u+\dfrac{\beta }{2u^3}. \end{aligned}$$

(8.4)

1.3 Painlevé III

The Painlevé III equation has the following rational form:

$$\begin{aligned} \dfrac{\mathrm{d}^2\lambda }{\mathrm{d}t^2}=\dfrac{1}{\lambda }\left( \dfrac{\mathrm{d}\lambda }{\mathrm{d}t}\right) ^2-\dfrac{1}{t}\dfrac{\mathrm{d}\lambda }{\mathrm{d}t}+\dfrac{1}{t}\left( \alpha \lambda ^2+\beta \right) +\gamma \lambda ^3+\dfrac{\delta }{\lambda }. \end{aligned}$$

(8.5)

In order to obtain the equivalent form of (8.5) which we use in Sect. 4, one can make the change of variables

$$\begin{aligned} \lambda \left( t\right) =\mathrm{e}^{u\left( t\right) }. \end{aligned}$$

Substituting $t=\mathrm{e}^{\tau }$ into (8.5), we get

$$\begin{aligned} \dfrac{\mathrm{d}^2 u}{\mathrm{d}t^2}=\alpha \mathrm{e}^{\tau +u}+\beta \mathrm{e}^{\tau -u}+\gamma \mathrm{e}^{2\left( \tau +u\right) }+\delta \mathrm{e}^{2\left( \tau -u\right) }. \end{aligned}$$

(8.6)

Appendix 2: Elliptic Functions

The definitions and properties of elliptic functions used in the paper can be found in [25, 36]. The main object is the theta function defined by

$$\begin{aligned} \theta \left[ \begin{array}{c}a\\ b\end{array}\right] \left( z,\tau \right) =\sum _{j\in \mathbb Z}q^{\frac{1}{2} (j+a)^2}\mathbf {e}\left( (j+a)(z+b)\right) , \end{aligned}$$

where $q=\mathbf {e}\left( \tau \right) \equiv \exp \left( 2\pi \mathrm{i}\tau \right)$.

We also use the Eisenstein functions

$$\begin{aligned} \varepsilon _k(z)= & {} \lim _{M\rightarrow +\infty }\sum _{n=-M}^M(z+n)^{-k}, \qquad k\in \mathbb {N},\nonumber \\ E_k(z)= & {} \lim _{M\rightarrow +\infty }\sum _{n=-M}^M\varepsilon _k(z+n\tau ). \end{aligned}$$

(9.1)

To determine limits of Lax matrices, we use the series expansions of the following functions:

$$\begin{aligned} \vartheta (z)= & {} \theta \left[ \begin{array}{l} 1/2 \\ 1/2 \end{array}\right] \left( z,\tau \right) =\sum _{j\in \mathbb {Z}}q^{\frac{1}{2}\left( j+\frac{1}{2}\right) ^2}\mathbf {e}\left( \left( j+\dfrac{1}{2}\right) \left( z+\dfrac{1}{2}\right) \right) ,\end{aligned}$$

(9.2)

$$\begin{aligned} \phi (u,z)= & {} \dfrac{\vartheta (u+z)\vartheta '(0)}{\vartheta (u)\vartheta (z)},\nonumber \\ \varphi _{\alpha }\left( u+\omega _{\beta },z\right)= & {} \mathbf {e}\left( z\partial _{\tau }\omega _{\alpha }\right) \phi \left( u+\omega _{\beta },z\right) ,\nonumber \\ f_{\alpha }\left( u+\omega _{\beta },z\right)= & {} \mathbf {e}\left( z\partial _{\tau }\omega _{\alpha }\right) \partial _w\phi \left( w,z\right) |_{w=u+\omega _{\beta }}, \end{aligned}$$

(9.3)

where $\omega _{\alpha }=\left\{ 0,\frac{1}{2},\frac{\tau }{2},\frac{1+\tau }{2}\right\}$. The functions satisfy the following well-known identities:

$$\begin{aligned} \phi (u,z)\phi (-u,z)= & {} E_2(z)-E_2(u),\nonumber \\ \partial _u\phi (u,z)= & {} \phi (u,z)(E_1(u+z)-E_1(u)), \end{aligned}$$

(9.4)

parity

$$\begin{aligned} E_k(-z)= & {} (-1)^k E_k(z),\\ \vartheta (-z)= & {} -\vartheta (z),\\ \phi (u,z)= & {} \phi (z,u)=-\phi (-u,-z), \end{aligned}$$

and quasi-periodicity

$$\begin{aligned} E_1(z+1)= & {} E_1(z),\quad E_1(z+\tau )=E_1(z)-2\pi \mathrm{i},\nonumber \\ E_2(z+1)= & {} E_2(z),\quad E_2(z+\tau )=E_2(z),\nonumber \\ \vartheta (z+1)= & {} -\vartheta (z),\quad \vartheta (z+\tau )=-q^{-\frac{1}{2}}\mathbf {e}(-z)\vartheta (z),\nonumber \\ \phi (u+1,z)= & {} \phi (u,z),\quad \phi (u+\tau ,z)=\mathbf {e}(-z)\phi (u,z). \end{aligned}$$

(9.5)

Using definition (9.3), we reduce the expansion of $\varphi _{\alpha }\left( u+\omega _{\beta },z\right)$ to the expansion of theta functions:

$$\begin{aligned} \phi (u-\sigma \tau ,z-\varsigma \tau )=\dfrac{\vartheta (u+z-(\sigma +\varsigma )\tau )\vartheta '(0)}{\vartheta (u-\sigma \tau )\vartheta (z-\varsigma \tau )}, \end{aligned}$$

and for the expansion of theta functions we have:

$$\begin{aligned} \vartheta \left( z+\sigma \tau \right)= & {} \left[ 1+\varvec{o}\left( 1\right) \right] q^{\left( -\left\lfloor \sigma \right\rfloor ^2/2+\frac{1}{8}-\left\lfloor \sigma \right\rfloor \left\{ \sigma \right\} -\left\{ \sigma \right\} /2\right) }\; \mathrm{e}\left( -\left\lfloor \sigma \right\rfloor z-\frac{\left\lfloor \sigma \right\rfloor }{2}\right) \\&\times \left\{ \begin{array}{ll} -2\sin \left( \pi z\right) ,&{}\left\{ \sigma \right\} =0,\\ -\mathrm{i}\;\mathrm{e}\left( -\dfrac{z}{2}\right) ,&{}\left\{ \sigma \right\} >0, \end{array}\right. \end{aligned}$$

where $\left\lfloor \sigma \right\rfloor$ is the integer part of $\sigma$ and $\left\{ \sigma \right\}$ is the fractional part of $\sigma$. This gives the following answer:

$$\begin{aligned}&\phi (u+\sigma _u\tau ,z+\sigma _z\tau )=\left( 1+\varvec{o}\left( 1\right) \right) \nonumber \\&\times \left\{ \begin{array}{llll} -2\pi \mathrm{i}q^{-\sigma _u\sigma _z+\left\{ \sigma _u\right\} \left\{ \sigma _z\right\} }\mathrm{e}\left( -\left\lfloor \sigma _z\right\rfloor u-\left\lfloor \sigma _u\right\rfloor z\right) ,&{} \left\{ \sigma _u\right\} >0,&{}\left\{ \sigma _z\right\} >0,&{}\left\{ \sigma _u\right\} +\left\{ \sigma _z\right\} <1,\nonumber \\ \begin{array}{l}4\pi q^{-\sigma _u\sigma _z+\left\{ \sigma _u\right\} \left\{ \sigma _z\right\} }\sin \left( \pi \left( u+z\right) \right) \\ \times \mathrm{e}\left( -\left( \frac{1}{2}+\left\lfloor \sigma _z\right\rfloor \right) u-\left( \frac{1}{2}+\left\lfloor \sigma _u\right\rfloor \right) z\right) \end{array},&{} \left\{ \sigma _u\right\} >0,&{}\left\{ \sigma _z\right\} >0,&{}\left\{ \sigma _u\right\} +\left\{ \sigma _z\right\} =1,\nonumber \\ \begin{array}{l}2\pi \mathrm{i}q^{-\sigma _u\sigma _z+\left\{ \sigma _u\right\} \left\{ \sigma _z\right\} -\left\{ \sigma _u+\sigma _z\right\} }\\ \times \mathrm{e}\left( -(1+\left\lfloor \sigma _z\right\rfloor )u-(1+\left\lfloor \sigma _u\right\rfloor )z\right) \end{array},&{} \left\{ \sigma _u\right\} >0,&{}\left\{ \sigma _z\right\} >0,&{}\left\{ \sigma _u\right\} +\left\{ \sigma _z\right\} >1,\\ \pi q^{-\sigma _u\sigma _z}\mathrm{e}\left( -\left\lfloor \sigma _z\right\rfloor u-\sigma _uz\right) \dfrac{\mathrm{e}\left( -\frac{u}{2}\right) }{\sin \left( \pi u\right) },&{} \left\{ \sigma _u\right\} =0,&{}\left\{ \sigma _z\right\} >0,\\ \pi q^{-\sigma _u\sigma _z}\mathrm{e}\left( -\sigma _zu-\left\lfloor \sigma _u\right\rfloor z\right) \dfrac{\mathrm{e}\left( -\frac{z}{2}\right) }{\sin \left( \pi z\right) },&{} \left\{ \sigma _u\right\} >0,&{}\left\{ \sigma _z\right\} =0,\\ \pi q^{-\sigma _u\sigma _z}\mathrm{e}\left( -\sigma _zu-\sigma _uz\right) \dfrac{\sin \left( \pi (u+z)\right) }{\sin \left( \pi z\right) \sin \left( \pi u\right) },&{} \left\{ \sigma _u\right\} =0,&{}\left\{ \sigma _z\right\} =0. \end{array}\right. \end{aligned}$$

To evaluate the limits of $f_{\alpha }\left( u+\omega _{\beta },z\right)$, we use the identity (9.4) and the expansion of $E_1(u-\sigma \tau )$ from [2]:

$$\begin{aligned} E_1(u-\sigma \tau )=2\pi \mathrm{i}\left\lfloor \sigma \right\rfloor +\left\{ \begin{array}{ll} \pi \cot (\pi u)+\varvec{\mathrm o}\left( 1\right) ,&{} \{\sigma \}=0,\\ \pi \mathrm{i}+2\pi \mathrm{i}q^{\{\sigma \}}\varvec{\mathrm{e}}(-u)+\varvec{\mathrm o}\left( q^{\{\sigma \}}\right) ,&{} 0<\{\sigma \}<\dfrac{1}{2},\\ \pi \mathrm{i}+2\pi \mathrm{i}q^{\frac{1}{2}}\left( \varvec{\mathrm{e}}(-u)-\varvec{\mathrm{e}}(u)\right) +\varvec{\mathrm o}\left( q^{\frac{1}{2}}\right) ,&{} \{\sigma \}=\dfrac{1}{2},\\ \pi \mathrm{i}-2\pi \mathrm{i}q^{1-\{\sigma \}}\varvec{\mathrm{e}}\left( u\right) +\varvec{\mathrm o}\left( q^{1-\{\sigma \}}\right) ,&{} \dfrac{1}{2}<\{\sigma \}<1. \end{array}\right. \end{aligned}$$