Connection formulae for the radial Toda equations I
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866916599013834752 |
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| author | Guest, Martin A. Its, Alexander R. Kosmakov, Maksim Miyahara, Kenta Odoi, Ryosuke |
| author_facet | Guest, Martin A. Its, Alexander R. Kosmakov, Maksim Miyahara, Kenta Odoi, Ryosuke |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2309_16550 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Connection formulae for the radial Toda equations I Guest, Martin A. Its, Alexander R. Kosmakov, Maksim Miyahara, Kenta Odoi, Ryosuke Mathematical Physics Classical Analysis and ODEs Exactly Solvable and Integrable Systems 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. |
| title | Connection formulae for the radial Toda equations I |
| topic | Mathematical Physics Classical Analysis and ODEs Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2309.16550 |