Solutions to autonomous partial difference equations via the third and sixth Painlevé equations and the Garnier system in two variables

Abstract

In this paper, we show that integrable autonomous partial difference equations admit special solutions described by the non-autonomous ordinary difference equations arising from the Bäcklund transformations of the third and sixth Painlevé equations and the Garnier system in two variables. Remarkably, although the equations themselves are autonomous, their special solutions are governed by non-autonomous ordinary difference equations.

Figures (3)

• Figure 3.1: Coxeter diagram describing the relations among the transformations $\{s^{(1)}_0,s^{(1)}_1,s^{(2)}_0,s^{(2)}_1,\pi^{(1)},\pi^{(2)},\pi^{(12)}\}$.
• Figure B.1: A cube for the system \ref{["eqns:CAC_PDE_ABCA'"]}. The $u$- and $v$-variables are assigned at the bottom and top vertices of the cube, respectively. Also, the equations \ref{["eqns:CAC_ABCA'"]} are assigned to the faces of the cube. For simplicity, we here use the shorthand notations: $u=u_{l,m}$, $v=v_{l,m}$, $\overline{\space}:\,l\to l+1$, $\widetilde{}\,:\,m\to m+1$.
• Figure C.1: A cube for the system \ref{['eqns:CABC_PDE_ASBC']}. The $u$- and $v$-variables are assigned at the bottom and top vertices of the cube, respectively. Also, each of the equations \ref{['eqns:CABC_ASBC']} is assigned to a face of the cube. Note that the face for Equation \ref{['eqn:CABC_S']} corresponds to the cutting plane bisecting the cube diagonally, and the faces for the equations \ref{['eqn:CABC_B']} and \ref{['eqn:CABC_BB']} correspond to the upper triangles. For simplicity, we here use the shorthand notations: $u=u_{l,m}$, $v=v_{l,m}$, $\overline{\space}:\,l\to l+1$, $\widetilde{}\,:\,m\to m+1$.

Theorems & Definitions (13)

• Remark 1.1
• Remark 1.2
• Theorem 2.1
• proof
• Theorem 2.2
• proof
• Theorem 2.3
• proof
• Remark 3.1
• Remark 3.2