Confluent Darboux transformations and Wronskians for algebraic solutions of the Painlevé III ($D_7$) equation
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arXiv
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| Format: | Preprint |
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2025
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| author | Harrow, J. W. E. Hone, A. N. W. |
| author_facet | Harrow, J. W. E. Hone, A. N. W. |
| contents | 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 $D_7$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_12696 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Confluent Darboux transformations and Wronskians for algebraic solutions of the Painlevé III ($D_7$) equation Harrow, J. W. E. Hone, A. N. W. Classical Analysis and ODEs Mathematical Physics Exactly Solvable and Integrable Systems 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 $D_7$. |
| title | Confluent Darboux transformations and Wronskians for algebraic solutions of the Painlevé III ($D_7$) equation |
| topic | Classical Analysis and ODEs Mathematical Physics Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2503.12696 |