Commutation Relations and Discrete Garnier Systems

Symmetry Integrability and Geometry-Methods and Applications

article

Christopher M. Ormerod¹  Eric M. Rains² 

[1] University of Maine, Department of Mathemaitcs & Statistics;California Institute of Technology

关键词: integrable systems;    dif ference equations;    Lax pairs;    discrete isomonodromy;   

DOI  :  10.3842/SIGMA.2016.110

来源: National Academy of Science of Ukraine

【 摘 要 】

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.

【 授权许可】

Unknown