Confluent Darboux transformations and Wronskians for algebraic solutions of the Painlev\'e III ($D_7$) equation
Year: 2025
Type: preprint
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 $D_7$.
Source: arXiv (Cornell University)
Authors: Harrow, J. W. E., Hone, A. N. W.
Language: English
Topic: Nonlinear Waves and Solitons
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W6979308508
Cites:
Cited by:
Related to: