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A new interpretation of Jimbo's formula for Painlevé VI

Zikang Wang, Yuancheng Xie, Xiaomeng Xu

Abstract

In this paper, we first give a new interpretation of Jimbo's boundary condition for the generic Painlevé VI transcendents, as the shrinking phenomenon in long time behaviour of the Jimbo-Miwa-Mori-Sato equation with rank $n=3$. We then interpret Jimbo's monodromy formula from the viewpoint of the isomonodromy deformation with respect to irregular singularities.

A new interpretation of Jimbo's formula for Painlevé VI

Abstract

In this paper, we first give a new interpretation of Jimbo's boundary condition for the generic Painlevé VI transcendents, as the shrinking phenomenon in long time behaviour of the Jimbo-Miwa-Mori-Sato equation with rank . We then interpret Jimbo's monodromy formula from the viewpoint of the isomonodromy deformation with respect to irregular singularities.
Paper Structure (18 sections, 15 theorems, 95 equations)

This paper contains 18 sections, 15 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. A new interpretation of Jimbo's formula for Painlevé VI
  3. A new interpretation of Jimbo's boundary condition
  4. A new interpretation of Jimbo's monodromy formula
  5. JMMS equations and Harnad duality
  6. Linear systems with one irregular singularity
  7. Isomonodromic deformation of the linear system \ref{['eq:irregularsystem']}
  8. Harnad Duality
  9. Painlevé VI equation and isomonodromic deformation
  10. PVI from isomonodromy deformation of Fuchsian system and Jimbo's formula
  11. Isomonodromy deformation of a linear system with irregular singularity
  12. The correspondence between the monodromy and solutions under the duality
  13. The correspondence of the boundary values and a new interpretation of Jimbo's formula
  14. The proof of Theorem \ref{['mainthm']}
  15. Asymptotic expansions of $k_1(x)$ and $k_2(x)$
  16. ...and 3 more sections

Key Result

Theorem 1.1

TangXu For almost every solution $\Phi(u)=\Phi_n(u)$ of the isomonodromy equation isoeq, there exist $n \times n$ matrix-valued functions $\Phi_{k}(u_1,...,u_k)$ for $k=1,...,n-1$ such that $\Phi_0:=\Phi_1$ is constant and for $2\leqslant k \leqslant n$ we have and where $\{\lambda^{(k-1)}_{i}\}_{i=1,\ldots,k-1}$ are the eigenvalues of the upper left $(k-1)\times (k-1)$ submatrix of $\Phi_{0}$.

Theorems & Definitions (29)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Corollary 1.6
  • Example 2.1
  • Remark 3.1
  • Theorem 3.2: Jimbo1982
  • Theorem 3.3: c.f. BalserLR
  • ...and 19 more