The non-linear steepest descent approach to the singular asymptotics of the sinh-Gordon reduction of the Painlevé III equation

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2025年1月13日

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Alexander R. Its,
Kenta Miyahara,
Maxim L. Yattselev

摘要

Motivated by the simplest case of tt*-Toda equations, we study the large and small x asymptotics for $ x>0 $ $x > 0$ of real solutions of the sinh-Godron Painlevé III( $D_6$ $D_{6}$ ) equation. These solutions are parametrized through the monodromy data of the corresponding Riemann–Hilbert problem. This unified approach provides connection formulae between the behavior at the origin and infinity of the considered solutions.