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The Degenerate Third Painleve' Equation: Complete Asymptotic Classification of Solutions in the Neighbourhood of the Regular Singular Point

A. V. Kitaev, A. Vartanian

TL;DR

This work provides a complete small-$\tau$ asymptotic classification for solutions of the degenerate third Painlevé equation in terms of the isomonodromic monodromy data of the associated 2×2 linear ODE. By leveraging the monodromy manifold, Backlund transformations, and complete asymptotic expansions, the authors derive generic and several special-case descriptions (power-like, logarithmic, and meromorphic) of the transcendent and its mole function, including detailed behavior near poles accumulating at the origin. They introduce branching parameters $\rho$ and $\varrho$ to parameterize local monodromy and provide explicit leading-term formulas with preserving symmetries, along with corrections and uniformizations via generating functions. The results deliver a unified framework linking monodromy data to diverse small-$\tau$ behaviours, including pole/zero structures and logarithmic regimes, paving the way for connection problems and $ au$-function analyses in future work.

Abstract

We give a classification for the small-$τ$ asymptotic behaviours of solutions to the degenerate third Painlevé equation, $u^{''}(τ) = \frac{(u^{\prime}(τ))^{2}}{u(τ)} - \frac{u^{\prime}(τ)}τ + \frac{1}τ\left(-8 \varepsilon (u(τ))^{2} + 2ab \right) + \frac{b^{2}}{u(τ)}, \quad\varepsilon=\pm1,\quad\varepsilon b>0, \quad a\in\mathbb{C}\setminus i\mathbb{Z}$, in terms of the monodromy data of a $2\times2$ matrix linear ODE whose isomonodromy deformations they describe. We also study the complete asymptotic expansions of the solutions.

The Degenerate Third Painleve' Equation: Complete Asymptotic Classification of Solutions in the Neighbourhood of the Regular Singular Point

TL;DR

This work provides a complete small- asymptotic classification for solutions of the degenerate third Painlevé equation in terms of the isomonodromic monodromy data of the associated 2×2 linear ODE. By leveraging the monodromy manifold, Backlund transformations, and complete asymptotic expansions, the authors derive generic and several special-case descriptions (power-like, logarithmic, and meromorphic) of the transcendent and its mole function, including detailed behavior near poles accumulating at the origin. They introduce branching parameters and to parameterize local monodromy and provide explicit leading-term formulas with preserving symmetries, along with corrections and uniformizations via generating functions. The results deliver a unified framework linking monodromy data to diverse small- behaviours, including pole/zero structures and logarithmic regimes, paving the way for connection problems and -function analyses in future work.

Abstract

We give a classification for the small- asymptotic behaviours of solutions to the degenerate third Painlevé equation, , in terms of the monodromy data of a matrix linear ODE whose isomonodromy deformations they describe. We also study the complete asymptotic expansions of the solutions.

Paper Structure

This paper contains 17 sections, 24 theorems, 225 equations.

Table of Contents

  1. Introduction
  2. The Monodromy Manifold and Bäcklund Transformations
  3. Small-$\tau$ Asymptotics: The Generic Case
  4. Power-Like Small-$\tau$ Asymptotics: The Special Cases $\rho\neq0$ and $\varrho\neq1/2$
  5. Solutions with Poles Accumulating at the Origin: $\mathrm{Re}(\varrho)=1/2$
  6. Special Solutions with Logarithmic Behaviour as $\tau\to0$: $\rho=0$ and $\varrho=1/2$
  7. Meromorphic Solutions
  8. Appendix. The Complete Small-$\tau$ Asymptotic Expansion of the General Solution $u(\tau)$
  9. Error Correction Term of the Power-Like Isomonodromy Asymptotics as $\tau\to0$
  10. Super-Generating Function
  11. A Symmetric Power-Like Asymptotic Expansion as $\tau\to0$
  12. Appendix. The Complete Small-$\tau$ Regular Logarithmic Asymptotic Expansion of $u(\tau)$: Theorem \ref{['th:ln-regular-a-non0']}
  13. Error Correction Term of the Isomonodromy Logarithmic Asymptotics As $\tau\to0$
  14. Super-Generating Function for the Regular Logarithmic Asymptotics
  15. Appendix. The Complete Small-$\tau$ Irregular Logarithmic Asymptotic Expansion of $u(\tau)$: Theorems \ref{['th:Asympt0-ln-rho12-']} and \ref{['th:Asympt0-ln-rho12+']}${}^{\mathbf\prime}$
  16. ...and 2 more sections

Key Result

Theorem 3.1

Let $(u(\tau), \varphi(\tau))$ be a solution of the system eq:dp3, eq:varphi corresponding to the monodromy data $(a,s_{0}^{0},s_{0}^{\infty},s_{1}^{\infty},g_{11},g_{12},g_{21},g_{22})$. Suppose that: (i) $s_0^\infty\neq0$, which implies that where either the upper signs or the lower signs, respectively, are taken;As discussed at the beginning of this section, the parameter $\varrho$ is defined

Theorems & Definitions (69)

  • Remark 2.1
  • Remark 3.1
  • Theorem 3.1
  • proof
  • Remark 3.2
  • Theorem 3.1
  • Remark 3.3
  • Proposition 3.1
  • proof
  • Corollary 3.1
  • ...and 59 more