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Connection formulae for the radial Toda equations II

Martin A. Guest, Alexander R. Its, Maksim Kosmakov, Kenta Miyahara, Ryosuke Odoi

TL;DR

The paper analyzes a monodromy partner of the radial Toda equation of type $A_2$, proving that real solutions $v_0(x)$ exist precisely on the $A^{\mathbb{R}}<0$ component of the monodromy manifold and obey a degenerate Painlevé III equation. Using a $3\times3$ Riemann-Hilbert framework, the authors employ a dressing strategy to overcome non-small-norm obstacles and extract the large-$x$ behavior, obtaining $v_0(x)=\frac{1}{2}\ln\left(\frac{1+\sin\vartheta(x)}{2-\sin\vartheta(x)}\right)+O(x^{-1})$ with a phase $\vartheta(x)$ depending on $(A^{\mathbb{R}},s^{\mathbb{R}},x^{\mathbb{R}},y^{\mathbb{R}})$ and $ u=-\frac{1}{2\pi i}\ln(3A^{\mathbb{R}})$. Through a change of variables, this yields large-$x$ asymptotics for the corresponding Painlevé III ($D_7$) equation, while the analysis also characterizes the movable singularities and shows nonexistence of real solutions on the opposite monodromy component. The results illuminate the link between connected components of the monodromy manifold and distinct differential equations, and the method lays groundwork for extending the RH approach to higher-rank cases $A_n$.

Abstract

This is a continuation of [arXiv:2309.16550] in which we computed the asymptotics near $x = \infty$ of all solutions of the radial Toda equation. In this article, we compute the asymptotics near $x = \infty$ of all solutions of a "partner" equation. The equations are related in the sense that their respective monodromy data constitute connected components of the same "monodromy manifold". While all solutions of the radial Toda equation are smooth, those of the partner equation have infinitely many singularities, and this makes the Riemann-Hilbert nonlinear steepest descent method (and the asymptotics of solutions) more involved.

Connection formulae for the radial Toda equations II

TL;DR

The paper analyzes a monodromy partner of the radial Toda equation of type , proving that real solutions exist precisely on the component of the monodromy manifold and obey a degenerate Painlevé III equation. Using a Riemann-Hilbert framework, the authors employ a dressing strategy to overcome non-small-norm obstacles and extract the large- behavior, obtaining with a phase depending on and . Through a change of variables, this yields large- asymptotics for the corresponding Painlevé III () equation, while the analysis also characterizes the movable singularities and shows nonexistence of real solutions on the opposite monodromy component. The results illuminate the link between connected components of the monodromy manifold and distinct differential equations, and the method lays groundwork for extending the RH approach to higher-rank cases .

Abstract

This is a continuation of [arXiv:2309.16550] in which we computed the asymptotics near of all solutions of the radial Toda equation. In this article, we compute the asymptotics near of all solutions of a "partner" equation. The equations are related in the sense that their respective monodromy data constitute connected components of the same "monodromy manifold". While all solutions of the radial Toda equation are smooth, those of the partner equation have infinitely many singularities, and this makes the Riemann-Hilbert nonlinear steepest descent method (and the asymptotics of solutions) more involved.

Paper Structure

This paper contains 17 sections, 5 theorems, 134 equations, 7 figures.

Table of Contents

  1. Introduction and Main Results
  2. Direct Monodromy Problem
  3. The Partner Equation
  4. More about the Monodromy Data
  5. Inverse Monodromy Problem and Asymptotics near $x = \infty$
  6. Riemann-Hilbert Problems
  7. Analogies and Differences in the Radial Toda and Partner Equations
  8. $\check{Y}$-problem
  9. Global Parametrix
  10. Local Parametrices
  11. $R$-problem
  12. $\Tilde{R}$-problem
  13. $\hat{R}$-problem
  14. Asymptotic Analysis
  15. Jump Matrices for the $\check{Y}$-problem
  16. ...and 2 more sections

Key Result

Theorem 1.1

The monodromy manifold corresponding to the solutions of radial Toda with x when n=2 is the connected component of $\mathcal{M}$ defined by monodromy cond with $A^\mathbb R>0$. The monodromy manifold corresponding to the solutions of third eq is the connected component of $\mathcal{M}$ defined by mo

Figures (7)

  • Figure 1: Monodromy manifold $\mathcal{M}$
  • Figure 2: Definition of the $\hat{\Psi}$-problem.
  • Figure 3: Contour $\Gamma_3$ and jump matrices of the $\check{Y}$-problem.
  • Figure 4: Arcs $C_n$, stationary points, and jump matrices of the $\check{Y}^D$-problem.
  • Figure 5: Regions $K^{(i)}$
  • ...and 2 more figures

Theorems & Definitions (10)

  • Theorem 1.1
  • Theorem 1.2
  • Remark
  • Corollary 1
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • Remark
  • Remark
  • Remark