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This paper has been published in Physica D. To download a preprint of this paper just click here.

This material is based upon work supported by the National Science Foundation under Grant Nos. DMS-1812625 and DMS-2204896 (Miller) and DMS-2108019 (Buckingham). Any opinions, findings and conclusions or recomendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation (NSF).

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On the Algebraic Solutions of the Painlevé-III (D7) Equation

Robert J. Buckingham and Peter D. Miller

RJB: Department of Mathematical Sciences, University of Cincinnati
PDM: Department of Mathematics, University of Michigan, Ann Arbor

Abstract:

The D7 degeneration of the Painlevé-III equation has solutions that are rational functions of \(x^{1/3}\) for certain parameter values. We apply the isomonodromy method to obtain a Riemann-Hilbert representation of these solutions. We demonstrate the utility of this representation by analyzing rigorously the behavior of the solutions in the large parameter limit.

Complex zeros of the Ohyama polynomials used to construct algebraic solutions of the Painlevé-III (D7) equation. Left: \(n=5\). Center: \(n=10\). Right: \(n=20\).