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The Boundary Condition for Some Isomonodromy Equations

Qian Tang, Xiaomeng Xu

Abstract

In this article, we study a special class of Jimbo-Miwa-Mori-Sato isomonodromy equations, which can be seen as a higher-dimensional generalization of Painlevé VI. We first construct its convergent $n\times n$ matrix series solutions satisfying certain boundary condition. We then use the Riemann-Hilbert approach to prove that the resulting solutions are almost all the solutions. Along the way, we find a shrinking phenomenon of the eigenvalues of the submatrices of the generic matrix solutions in the long time behaviour.

The Boundary Condition for Some Isomonodromy Equations

Abstract

In this article, we study a special class of Jimbo-Miwa-Mori-Sato isomonodromy equations, which can be seen as a higher-dimensional generalization of Painlevé VI. We first construct its convergent matrix series solutions satisfying certain boundary condition. We then use the Riemann-Hilbert approach to prove that the resulting solutions are almost all the solutions. Along the way, we find a shrinking phenomenon of the eigenvalues of the submatrices of the generic matrix solutions in the long time behaviour.
Paper Structure (27 sections, 49 theorems, 275 equations, 1 figure)

This paper contains 27 sections, 49 theorems, 275 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main result
  3. The idea of Proof: the Riemann-Hilbert method
  4. A new interpretatioin of Jimbo's formula for Painlevé VI
  5. Further directions: the special solutions
  6. Series Solutions and the Shrinking Solution
  7. Construction of Formal Series Solutions
  8. Convergence of the Formal Series Solutions
  9. Definition of the Shrinking Solutions
  10. Uniqueness of shrinking solutions
  11. Solution with Shrinking Phenomenon
  12. Effective Criterion from Asymptotic Behavior
  13. Explicit Expressions of Monodromy Data
  14. Confluent Hypergeometric Systems
  15. Stokes matrices
  16. ...and 12 more sections

Key Result

Theorem 1.1

For almost all the solutions $\Phi(u)=\Phi_n(u)$ of the $n$-th isomonodromy equation isoeq, there exists a sequence of $n \times n$ matrix-valued functions $\Phi_k(u_1,\ldots,u_k)$ for $k=1,\ldots,n$ and a constant $n\times n$ matrix $\Phi_0$ such that for every $1\leqslant k \leqslant n$ and in the meanwhile $\Phi_0$ satisfies the boundary condition where $\{\lambda^{(k-1)}_{i}\}_{i=1,\ldots,k-1

Figures (1)

  • Figure 1: Numerical simulation of the shrinking phenomenon when $n=3$

Theorems & Definitions (76)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Example 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.7
  • Proposition 1.8
  • Remark 2.1
  • Theorem 2.2
  • ...and 66 more