Connection formulae for the radial Toda equations I

TL;DR

This paper develops connection formulae for the radial solutions of the 2D periodic Toda equations, combining PDE analysis and Riemann-Hilbert methods to understand asymptotic behavior at zero and infinity.

Contribution

It introduces a novel approach to derive connection formulae for the 2D Toda equations using advanced integrability techniques, addressing the case n=2.

Findings

Established connection formulae for n=2 case

Extended nonlinear steepest descent analysis to 3x3 Riemann-Hilbert problems

Captured key features applicable to general n cases

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.