Confluent Darboux transformations and Wronskians for algebraic solutions of the Painlevé III (D₇) equation

2025/03/16 by Harrow, J. W. E., Hone, A. N. W.
#Classical Analysis and ODEs (math.CA) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2503.12696

Abstract

We describe the use of confluent Darboux transformations for Schrödinger operators, and how they give rise to explicit Wronskian formulae for certain algebraic solutions of Painlevé 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 Painlevé 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 Painlevé III equation of type D₇.

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