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Painlevé IV with both Parameters Zero: A Numerical Study


  • Jonah A. Reeger,

    1. University of Colorado
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    • Captain, United States Air Force, supported by the Department of Defense. The views expressed in this article are those of the authors and do not reflect the official policy or position of the United States Air Force, Department of Defense, or U.S. Government.

  • Bengt Fornberg

    1. University of Colorado
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    • Supported by NSF grant DMS-0914647.

Jonah A. Reeger, University of Colorado, Department of Applied Mathematics, 526 UCB, Boulder, CO 80309, USA; e-mail:


The six Painlevé equations were introduced over a century ago, motivated by rather theoretical considerations. Over the last several decades, these equations and their solutions, known as the Painlevé transcendents, have been found to play an increasingly central role in numerous areas of mathematical physics. Due to extensive dense pole fields in the complex plane, their numerical evaluation remained challenging until the recent introduction of a fast “pole field solver” [1]. The fourth Painlevé equation has two free parameters in its coefficients, as well as two free initial conditions. The present study applies this new computational tool to the special case when both of its parameters are zero. We confirm existing analytic and asymptotic knowledge about the equation, and also explore solution regimes which have not been described in the previous literature.

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