Presented By: Integrable Systems and Random Matrix Theory Seminar - Department of Mathematics
Transition Asymptotics for the Real Solutions of the sinh-Gordon Painlev\'e III
Kenta Miyahara (Indiana University Indianapolis)
We consider solutions of the sinh-Gordon Painlev\'e III equation
\[
u_{xx} + \frac{1}{x} u_x = \sinh u
\]
that are real on \((0, \infty)\). They are parametrized by the monodromy parameter \( p \in \overline{\mathbb C} \), \( |p|>1 \), and an additional real parameter \( s^{\mathbb R} \) when \( p=\infty \). We describe the transition between singular solutions (\( |p|<\infty \)) and smooth solutions (\( p=\infty \)), as \( x \to +\infty \) and \( p \to \infty \) given that \( 2\Im(p)=-s^\R\).
This presentation is based on the ongoing work with Maxim Yattselev.
\[
u_{xx} + \frac{1}{x} u_x = \sinh u
\]
that are real on \((0, \infty)\). They are parametrized by the monodromy parameter \( p \in \overline{\mathbb C} \), \( |p|>1 \), and an additional real parameter \( s^{\mathbb R} \) when \( p=\infty \). We describe the transition between singular solutions (\( |p|<\infty \)) and smooth solutions (\( p=\infty \)), as \( x \to +\infty \) and \( p \to \infty \) given that \( 2\Im(p)=-s^\R\).
This presentation is based on the ongoing work with Maxim Yattselev.
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