Padé Interpolation for Elliptic Painlevé Equation

Abstract

An interpolation problem related to the elliptic Painlevé equation is formulated and solved. A simple form of the elliptic Painlevé equation and the Lax pair are obtained. Explicit determinant formulae of special solutions are also given.

Appendix: Affine Weyl Group Actions

Here we give a derivation of the Painlevé equation (39), (40) from the affine Weyl group actions [9, 21].

Define multiplicative transformations s _ij , c, μ _ij , ν _ij (1≤i≠j≤8) acting on variables h ₁,h ₂,u ₁,…,u ₈ as

(71)

These actions generate the affine Weyl group of type $E^{(1)}_{8}$ with the following simple reflections:

$$ \begin{array}{cccccccccccccccccc} &&&&s_{12}\\ &&&&\vert\\ c&-&\mu_{12}&-&s_{23}&-&s_{34}&-&\cdots&-&s_{78} . \end{array} $$

(72)

We extend the actions bi-rationally on variables (f,g). The nontrivial actions are as follows:

$$ c(f)=g, \qquad c(g)=f, \qquad \mu_{ij}(f)=\tilde{f},\qquad \nu_{ij}(g)=\tilde{g}, $$

(73)

where, $\tilde{f}=\tilde{f}_{ij}$ and $\tilde{g}=\tilde{g}_{ij}$ are rational functions in (f,g) defined by

$$ \frac{\tilde{f}-\mu_{ij}(f_i)}{\tilde{f}-\mu_{ij}(f_j)}=\frac {(f-f_i)(g-g_j)}{(f-f_j)(g-g_i)},\qquad \frac{\tilde{g}-\nu_{ij}(g_i)}{\tilde{g}-\nu_{ij}(g_j)}= \frac {(g-g_i)(f-f_j)}{(g-g_j)(f-f_i)}, $$

(74)

(f _i ,g _i )=(f _⋆(u _i ),g _⋆(u _i )), and

$$ f_\star(z)=\frac{\vartheta_p(\frac {d_2}{z},\frac {h_1}{d_2 z})}{\vartheta_p(\frac {d_1}{z},\frac {h_1}{d_1 z})},\qquad g_\star(z)= \frac{\vartheta_p(\frac {d_2}{z},\frac {h_2}{d_2 z})}{\vartheta_p(\frac {d_1}{z},\frac {h_2}{d_1 z})}, $$

(75)

as in Eq. (10). As a rational function of (f,g), $\tilde{f}$ is characterized by the following properties: (i) it is of degree (1,1) with indeterminate points (f _i ,g _i ), (f _j ,g _j ), (ii) it maps generic points on the elliptic curve (f _⋆(z),g _⋆(z)) to $\frac{\vartheta_{p}(\frac {d_{2}}{z},\frac {h_{1}h_{2}}{d_{2} z u_{1} u_{2}})}{\vartheta_{p}(\frac {d_{1}}{z},\frac {h_{1}h_{2}}{d_{1} z u_{i} u_{j}})}$. Using this geometric characterization, we have

$$ \mu_{ij} \biggl\{\frac{{\mathcal{F}}_f(\frac {h_1 z}{h_2})}{{\mathcal{F}}_f(z)} \biggr\}= \frac{\vartheta_p(\frac {u_i}{z},\frac {u_j}{z})}{\vartheta_p(\frac {h_2}{u_i z},\frac {h_2}{u_j z})} \frac{{\mathcal{F}}_f(\frac {h_1 z}{h_2})}{{\mathcal{F}}_f(z)}, \quad \mbox{for } g=g_\star(z), $$

(76)

where the functions ${\mathcal{F}}_{f}(z)$ (and ${\mathcal{G}}_{g}(z)$) are defined in a similar way as Eq. (11)

(77)

Let us consider the following compositions [9]

$$ r=s_{12}\mu_{12}s_{34}\mu_{34}s_{56} \mu_{56}s_{78}\mu_{78}, \qquad T=rcrc. $$

(78)

Their actions on variables (h _i ,u _i ) are given by

(79)

where v=qh ₂/h ₁, $q=h_{1}^{2}h_{2}^{2}/(u_{1}\cdots u_{8})$. From Eq. (77) and $r(\frac {h_{1}}{h_{2}})=\frac {q h_{2}}{h_{1}}$, the evolution T(f)=rcrc(f)=r(f) is determined as

$$ \frac{{\mathcal{F}}_f(z)}{{\mathcal{F}}_f(\frac {h_1 z}{h_2})}\frac{T({\mathcal{F}}_f)(\frac {q h_2 z}{h_1})}{T({\mathcal{F}}_f)(z)} =\prod_{i=1}^8 \frac{\vartheta_p(\frac {u_i}{z})}{\vartheta_p(\frac {h_2}{u_i z})}, \quad \mbox{for } g=g_\star(z). $$

(80)

Similarly, since cTc=T ⁻¹, T ⁻¹(g) is determined by

$$ \frac{{\mathcal{G}}_g(z)}{{\mathcal{G}}_g(\frac {h_2 z}{h_1})}\frac{T^{-1}({\mathcal{G}}_g)(\frac {q h_1 z}{h_2})}{T^{-1}({\mathcal{G}}_g)(z)} =\prod_{i=1}^8 \frac{\vartheta_p(\frac {u_i}{z})}{\vartheta_p(\frac {h_1}{u_i z})}, \quad \mbox{for } f=f_\star(z). $$

(81)

By a re-scaling of variables (h _i ,u _i ,d _i )=(κ _i λ ²,ξ _i λ,c _i λ) with $\lambda=(h_{1}^{3} h_{2}^{-1})^{\frac{1}{4}}$, we have ${\mathcal{F}}_{f}(z)=F_{f}(\frac {z}{\lambda})$, $T({\mathcal{F}}_{f})(z)=T(F_{f})(\frac {\kappa_{1}}{\kappa_{2}}\frac {z}{\lambda})$ and so on, since $T(\lambda)=\frac {h_{2}}{h_{1}}\lambda$. Then the above equations take the form (39), (40), by putting z=λx.