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Regular flat structure and generalized Okubo system. (English) Zbl 1444.34109

A matrix system of differential equations of the form \[ (z-T)\frac{dY}{dz}=-B_{\infty} Y \] on \(\mathbb{P}^1\) with constant \(N \times N\) matrices \(T\) and \(B_{\infty}\) is called an Okubo system if \(T\) is diagonalizable, and a generalized Okubo system if \(T\) is not necessarily diagonalizable. Let \(U\) be a simply-connected domain in \(\mathbb{C}^N\). Suppose that each entry of \(T\) is holomorphic on \(U\), that \(T \sim J_1 \oplus \cdots \oplus J_n,\) and that \(B_{\infty}=\text{diag}(\lambda_1, \dots,\lambda_N)\), where \(J_j\)’s are of Jordan normal form and \(\lambda_j \in\mathbb{C}\), \(\lambda_i-\lambda_j \not\in \mathbb{Z}\setminus \{ 0\}\) if \(i\not=j\). By adding a suitable matrix valued \(1\)-form \(\tilde{\Omega}\) on \(U\), this is extended to the completely integrable Pfaffian system \[ dY= -(zI_N-T)^{-1} (dz+ \tilde{\Omega}) B_{\infty} Y \] on \(\mathbb{P}^1 \times U\), which is called an extended generalized Okubo system. In the paper under review, it is shown that an extended generalized Okubo system gives isomonodromic deformation of the generalized Okubo system, and that, under a certain condition, the space of deformation parameters of an extended generalized Okubo system may be equipped with a flat structure, namely, a Saito structure. By using the result of H. Kawakami [Int. Math. Res. Not. 2010, No. 17, 3394–3421 (2010; Zbl 1207.34119)] saying that any system of linear differential equations is transformed into a generalized Okubo system, the authors find extended generalized Okubo systems governed by five Painlevé equations PII, \(\dots,\) PVI, respectively. Consequently, flat structures are introduced on the spaces of variables of generic solutions to these Painlevé equations, and then PII, \(\dots,\) PVI are treated uniformly as the four-dimensional extended WDVV equation. Furthermore the well-known coalescence cascade of the Painlevé equations is understood to be the degeneration scheme of the Jordan normal forms of a square matrix of rank four.

MSC:

34M56 Isomonodromic deformations for ordinary differential equations in the complex domain
34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
58A17 Pfaffian systems

Citations:

Zbl 1207.34119

References:

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