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The non-linear steepest descent approach to the singular asymptotics of the sinh-Gordon reduction of the Painlevé III equation

Alexander R. Its, Kenta Miyahara, Maxim L. Yattselev

Abstract

Motivated by the simplest case of tt*-Toda equations, we study the large and small $x$ asymptotics for $x>0$ of real solutions of the sinh-Godron Painlevé III($D_6$) equation. These solutions are parametrized through the monodromy data of the corresponding Riemann-Hilbert problem. This unified approach provides connection formulae between the behavior at the origin and infinity of the considered solutions.

The non-linear steepest descent approach to the singular asymptotics of the sinh-Gordon reduction of the Painlevé III equation

Abstract

Motivated by the simplest case of tt*-Toda equations, we study the large and small asymptotics for of real solutions of the sinh-Godron Painlevé III() equation. These solutions are parametrized through the monodromy data of the corresponding Riemann-Hilbert problem. This unified approach provides connection formulae between the behavior at the origin and infinity of the considered solutions.

Paper Structure

This paper contains 17 sections, 5 theorems, 136 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Main Results
  3. Proof of Theorem \ref{['Niles result']}
  4. Direct Monodromy Problem for sinh-Gordon Equation
  5. Riemann-Hilbert Problem for sinh-Gordon Equation
  6. Riemann-Hilbert Problem for sine-Gordon Equation
  7. Proof of Theorem \ref{['main thm']}
  8. Case 1: $B^{\mathbb R} = 0$
  9. Case 2: $B^{\mathbb R} \neq 0$
  10. Opening of Lenses
  11. The Global Parametrix
  12. Local Parametrix at $\lambda = 1$
  13. Local Parametrix at $\lambda = -1$
  14. Small Norm Problem
  15. Asymptotic Analysis
  16. ...and 2 more sections

Key Result

Theorem 2.1

To each finite $p$, $|p|>1$, there corresponds one real solution of the sinh-Gordon Painlevé III equation sinh-gordon PIII on $\mathbb R_{>0}$ and as $x \to +\infty$, where the error term is uniform in $x$, To $p=\infty$ there corresponds a one-parameter family of real solutions of the sinh-Gordon Painlevé III equation sinh-gordon PIII on $\mathbb R_{>0}$ such that as $x\to+\infty$, where $s^\m

Figures (13)

  • Figure 1: Monodromy surface $\mathcal{M}$ for the sinh-Gordon Painlevé III equation.
  • Figure 2: The contour $\Gamma$ and the jump matrices $G_{\hat{\Psi}}$ for RHP \ref{['rhp original']}.
  • Figure 3: The jump matrices $G_{\hat{\Phi}}$ for RHP \ref{['rhp niles']}.
  • Figure 4: The jump of $Y$ on the imaginary line $\Gamma_Y$.
  • Figure 5: The jump matrices $G_{X}$ for RHP \ref{['RHP X']}.
  • ...and 8 more figures

Theorems & Definitions (17)

  • Theorem 2.1
  • Remark
  • Remark
  • Remark
  • Remark
  • Theorem 2.2
  • Remark
  • Remark
  • Remark
  • Remark
  • ...and 7 more