Confluent Darboux transformations and Wronskians for algebraic solutions of the Painlev\'e III ($D_7$) equation
Classical Analysis and ODEs
2025-03-18 v1 Mathematical Physics
math.MP
Exactly Solvable and Integrable Systems
Abstract
We describe the use of confluent Darboux transformations for Schr\"odinger operators, and how they give rise to explicit Wronskian formulae for certain algebraic solutions of Painlev\'e equations. As a preliminary illustration, we briefly describe how the Yablonskii-Vorob'ev polynomials arise in this way, thus providing well-known expressions for the tau functions of the rational solutions of the Painlev\'e II equation. We then proceed to apply the method to obtain the main result, namely a new Wronskian representation for the Ohyama polynomials, which correspond to the algebraic solutions of the Painlev\'e III equation of type .
Cite
@article{arxiv.2503.12696,
title = {Confluent Darboux transformations and Wronskians for algebraic solutions of the Painlev\'e III ($D_7$) equation},
author = {J. W. E. Harrow and A. N. W. Hone},
journal= {arXiv preprint arXiv:2503.12696},
year = {2025}
}