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Commutation Relations and Discrete Garnier Systems

Ormerod, Christopher M. and Rains, Eric M. (2016) Commutation Relations and Discrete Garnier Systems. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA), 12 . Art. No. 110. ISSN 1815-0659. doi:10.3842/SIGMA.2016.110.

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We present four classes of nonlinear systems which may be considered discrete analogues of the Garnier system. These systems arise as discrete isomonodromic deformations of systems of linear difference equations in which the associated Lax matrices are presented in a factored form. A system of discrete isomonodromic deformations is completely determined by commutation relations between the factors. We also reparameterize these systems in terms of the image and kernel vectors at singular points to obtain a separate birational form. A distinguishing feature of this study is the presence of a symmetry condition on the associated linear problems that only appears as a necessary feature of the Lax pairs for the least degenerate discrete Painlevé equations.

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Additional Information:The authors retain the copyright for their papers published in SIGMA under the terms of the Creative Commons Attribution-ShareAlike License. Received March 30, 2016, in final form October 30, 2016; Published online November 08, 2016. The work of EMR was partially supported by the National Science Foundation under the grant DMS-1500806.
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Subject Keywords:integrable systems; difference equations; Lax pairs; discrete isomonodromy
Classification Code:2010 Mathematics Subject Classification: 39A10; 39A13; 37K15
Record Number:CaltechAUTHORS:20161215-111704158
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:72849
Deposited By: Tony Diaz
Deposited On:15 Dec 2016 19:31
Last Modified:11 Nov 2021 05:07

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