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arXiv · 2309.16550

Connection formulae for the radial Toda equations I

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

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Martin A. Guest, Alexander R. Its, Maksim Kosmakov, Kenta Miyahara, Ryosuke Odoi. 2023-09-28. Connection formulae for the radial Toda equations I. https://arxiv.org/abs/2309.16550

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