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Asymptotic and monodromy problems for higher-order Painlevé III equations

Zikang Wang, Xiaomeng Xu

TL;DR

This work generalizes isomonodromy deformations to $n\times n$ meromorphic systems with two irregular poles, formulating a boundary-value parameterization of solutions via asymptotic data on $T^*GL_n$. It provides explicit monodromy data formulas (Stokes and connection matrices) in terms of these asymptotic parameters and establishes a Riemann–Hilbert-type decomposition that reduces the two-pole problem to one-pole subsystems. The authors construct shrinking solutions that admit precise limits as $t\to0$ and show that almost all solutions are shrinking, while non-shrinking cases form an open dense but proper subset linked to tt^{*}$–theory. Specializations to sine-Gordon Painlevé III and tt^{*}-Toda equations recover known results and illuminate the geometric role of the asymptotic data via connections to crystal-like structures and polyhedral parameterizations. Overall, the paper provides a unified isomonodromy framework for high-order Painlevé-III-type systems with two irregular singularities and demonstrates its relevance to tt^{*}$ equations and related integrable structures.

Abstract

In this paper, we study the isomonodromy deformation equations for the $n\times n$ system of first order meromorphic linear ordinary differential equations with two second order poles. We analyze the asymptotic behaviour of the solutions at a boundary point of the isomonodromic deformation space, and derive a parameterization of the solutions via asymptotic parameters. We then derive the explicit formula for the Stokes matrices and connection matrix of the associated linear system in terms of the asymptotic parameters. In the end, we apply the results to the study of the $tt^{*}$ equations.

Asymptotic and monodromy problems for higher-order Painlevé III equations

TL;DR

This work generalizes isomonodromy deformations to meromorphic systems with two irregular poles, formulating a boundary-value parameterization of solutions via asymptotic data on . It provides explicit monodromy data formulas (Stokes and connection matrices) in terms of these asymptotic parameters and establishes a Riemann–Hilbert-type decomposition that reduces the two-pole problem to one-pole subsystems. The authors construct shrinking solutions that admit precise limits as and show that almost all solutions are shrinking, while non-shrinking cases form an open dense but proper subset linked to tt^{*} equations and related integrable structures.

Abstract

In this paper, we study the isomonodromy deformation equations for the system of first order meromorphic linear ordinary differential equations with two second order poles. We analyze the asymptotic behaviour of the solutions at a boundary point of the isomonodromic deformation space, and derive a parameterization of the solutions via asymptotic parameters. We then derive the explicit formula for the Stokes matrices and connection matrix of the associated linear system in terms of the asymptotic parameters. In the end, we apply the results to the study of the equations.
Paper Structure (22 sections, 35 theorems, 215 equations, 2 figures)

This paper contains 22 sections, 35 theorems, 215 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Asymptotics of solutions of isomonodromy deformation equations
  3. The monodromy problem for the meromorphic linear ODEs
  4. Connection to the $tt^{*}$ equations and examples of solutions not in $\mathfrak{Sol}_{Shr}$
  5. Preliminaries on the meromorphic linear systems with one second order pole
  6. Long time behaviour and parameterization of the generic solutions of the equation \ref{['eq:iso eq of one irr in u']}
  7. Monodromy matrices of the linear ordinary differential equation
  8. The isomonodromy property
  9. The isomonodromy deformation and monodromy problem
  10. System with two irregular singularities, its monodromy data and isomonodromy equation
  11. Lax pair of the nonlinear PDEs \ref{['eq:iso eq of two irr A-u']}-\ref{['eq:iso eq of two irr G-v']}
  12. Monodromy data
  13. The isomonodromy property
  14. Construction of solutions of the isomonodromy equations with the prescribed asymptotics as $t\rightarrow 0$
  15. Monodromy of the linear system with 2 irregular singularities and Riemann-Hilbert correspondence
  16. ...and 7 more sections

Key Result

Theorem 1.1

For any $( G_0,\widehat{A}_0)\in {\rm GL}_n\times \frak{gl}_n$ satisfying the boundary condition, there exists a unique multi-valued meromorphic solution $A(\mathbf{z},t,\mathbf{w};\widehat{A}_0,G_0),G(\mathbf{z},t,\mathbf{w};\widehat{A}_0,G_0)$ of the isomonodromy equations iso for z begin–iso for and Here in the last identity, we let $z_{n-1},...,z_2$ and $w_{n-1},...,w_2$ tend to infinity suc

Figures (2)

  • Figure 1: Discrimination of Monodromy matrix $M=\nu^{(\infty)}_{\frac{\pi}{8}}$ for order 4
  • Figure 2: Discrimination of Monodromy matrix $M=\nu^{{(\infty)}}_{\frac{\pi}{10}}$ for order 5

Theorems & Definitions (72)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 2.1: TangXu
  • Remark 2.2
  • Definition 2.3
  • Theorem 2.4
  • Definition 2.5
  • Lemma 2.6
  • ...and 62 more