Commutation Relations and Discrete Garnier Systems

Abstract

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.

References in corpus (4)

• A Lax Formalism for the Elliptic Difference Painlevé Equation
• Construction of a Lax Pair for the $E_6^{(1)}$ $q$-Painlevé System
• The Lattice Structure of Connection Preserving Deformations for q-Painlevé Equations I
• Symmetries in Connection Preserving Deformations

Cited by in corpus (6)

• Degenerations of Ruijsenaars-van Diejen operator and q-Painleve equations
• Variations of $q$-Garnier system
• An Elliptic Garnier System from Interpolation
• An elliptic Garnier system
• A Variation of the $q$-Painlevé System with Affine Weyl Group Symmetry of Type $E_7^{(1)}$
• Four-dimensional Painlevé-type difference equations