Connection formulae for the radial Toda equations I

TL;DR

The paper solves the connection problem for the radial 2D periodic Toda equation of type $A_2$ with $\epsilon=+1$ by extending the Deift–Zhou nonlinear steepest descent to a rank-3 matrix Riemann–Hilbert problem. It builds a complete inverse monodromy framework (Lax pair, monodromy data, and symmetry/reality constraints) and then performs a sequence of RH problem deformations, global and local parametrices, and a small-norm analysis to extract large-$x$ asymptotics for $w_0(x)$. The main result is explicit connection formulae that express the infinity oscillation parameters $(\sigma,\psi)$ in terms of the zero-data $(\gamma,\rho)$ via monodromy data, as well as a reciprocal description using $q^{\mathbb R}$ and $\gamma$. This advances the understanding of the global behavior of radial Toda solutions and provides a blueprint for extending the method to general $n$ in subsequent work, with potential applications to tt*-Toda and Painlevé III dynamics.

Abstract

This paper is the first in a forthcoming series of works where the authors study the global asymptotic behavior of the radial solutions of the 2D periodic Toda equation of type $A_n$. The principal issue is the connection formulae between the asymptotic parameters describing the behavior of the general solution at zero and infinity. To reach this goal we are using a fusion of the PDE analysis and the Riemann-Hilbert nonlinear steepest descent method of Deift and Zhou which is applicable to 2D Toda in view of its Lax integrability. A principal technical challenge is the extension of the nonlinear steepest descent analysis to Riemann-Hilbert problems of matrix rank greater than $2$. In this paper, we meet this challenge for the case $n=2$ (the rank $3$ case) and it already captures the principal features of the general $n$ case.

Figures (16)

• Figure 1: Stokes rays and Stokes sectors at $\zeta = 0$. (Here we only depicted $\Omega_1^{(0)}$ and $\Omega_2^{(0)}$, but the others are analogous.)
• Figure 2: Stokes rays and Stokes sectors at $\zeta = \infty$.
• Figure 3: Riemann-Hilbert Problem of $\hat{\Psi}$ (original domains of $\Psi$ shown, for clarity).
• Figure 4: Contour $\Gamma_2$ and jump matrix $G_\Phi$ of $\Phi$.
• Figure 5: Contour $\Gamma_2$ and jump matrix $G_{\tilde{\Phi}}$ of $\tilde{\Phi}$.

Theorems & Definitions (57)

• Theorem 1.1
• Proposition 2.1
• proof
• Remark
• Remark
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4