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Kapaev's global asymptotics of the fourth Painlevé transcendents. Elliptic asymptotics

Shun Shimomura

Abstract

For the fourth Painlevé transcendents we derive elliptic asymptotic representations, which were announced by late Professor Kapaev without proofs. Then we newly obtain related results including the correction function.

Kapaev's global asymptotics of the fourth Painlevé transcendents. Elliptic asymptotics

Abstract

For the fourth Painlevé transcendents we derive elliptic asymptotic representations, which were announced by late Professor Kapaev without proofs. Then we newly obtain related results including the correction function.

Paper Structure

This paper contains 25 sections, 205 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Results
  3. Solutions for $0<|\phi|<\pi/4$ in Kapaev-3
  4. Alternative expression of solutions
  5. Observation related to trigonometric asymptotics
  6. Basic facts
  7. Monodromy data
  8. Isomonodromy linear system
  9. Turning points, Stokes graph, WKB analysis
  10. Direct monodromy problem
  11. Asymptotics of monodromy data
  12. Integrals
  13. Theta-function
  14. Expression of $B_{\phi}(t)$
  15. Proofs of the main results
  16. ...and 10 more sections

Figures (11)

  • Figure 2.1: Cycles $\mathbf{a},$$\mathbf{b}$ on $\Pi_{A,\phi}=\Pi_+\cup\Pi_-$
  • Figure 4.1: Limit Stokes graphs on $\mathcal{R}_{\infty}^+$
  • Figure 5.1: Fragments of the Stokes graph for $-\pi/4<\phi<0$
  • Figure 5.2: Fragments of the Stokes graph for $0<\phi<\pi/4$
  • Figure 7.1: Cycle $\mathbf{b}=\mathbf{l}_{-1}\cup\mathbf{l}_{-2} \cup \mathbf{l}_2 \cup\mathbf{l}_1$
  • ...and 6 more figures

Theorems & Definitions (15)

  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • ...and 5 more