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An Elliptic Garnier System

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Abstract

We present a linear system of difference equations whose entries are expressed in terms of theta functions. This linear system is singular at \({4m+12}\) points for \({m \geq 1}\), which appear in pairs due to a symmetry condition. We parameterize this linear system in terms of a set of kernels at the singular points. We regard the system of discrete isomonodromic deformations as an elliptic analogue of the Garnier system. We identify the special case in which m = 1 with the elliptic Painlevé equation; hence, this work provides an explicit form and Lax pair for the elliptic Painlevé equation.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Maine, 5752 Neville Hall, Room 322, Orono, ME, 04469, USA

    Chris M. Ormerod

  2. Caltech, 1200 E California Blvd, Pasadena, CA, 91125, USA

    Eric M. Rains

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  1. Chris M. Ormerod
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  2. Eric M. Rains
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Correspondence to Chris M. Ormerod.

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Communicated by A. Borodin

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Ormerod, C.M., Rains, E.M. An Elliptic Garnier System. Commun. Math. Phys. 355, 741–766 (2017). https://doi.org/10.1007/s00220-017-2934-6

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  • DOI: https://doi.org/10.1007/s00220-017-2934-6

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