An Elliptic Garnier System

2016/07/27 by Christopher M. Ormerod, Chris M. Ormerod, Eric M. Rains · 7 citations
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Computer science #Elliptic curve #Elliptic function #Elliptic rational functions #Fractional Differential Equations Solutions #Geometry #Jacobi elliptic functions #Mathematical analysis #Mathematics #Nonlinear Waves and Solitons #Physics #Pure mathematics #Quarter period #Set (abstract data type) #Supersingular elliptic curve #Symmetry (geometry) #Work (physics) #math-ph #math.MP #msc:14H70 #msc:14K25 #msc:39A06 #nlin.SI

paper · pdf · doi:10.1007/s00220-017-2934-6

published in Communications in Mathematical Physics 355(2), 741-766 (Springer Science+Business Media) · 27 pages, 2 figures

arxiv created 2016/07/27 · openalex publication_date 2017/07/24 · arxiv updated 2017/09/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

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 ≥ 1, which appear in pairs due to a symmetry condition. We parameterize this linear system in terms 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.

Citations

• Padé interpolation for elliptic Painlevé equation
• A q-anaolg of the sixth Painlevé equation
• Analytic theory of difference equations with rational and elliptic coefficients and the Riemann-Hilbert problem
• The Lattice Structure of Connection Preserving Deformations for q-Painlevé Equations I
• The noncommutative geometry of elliptic difference equations

Cited by

• A Variation of the q-Painlevé System with Affine Weyl Group Symmetry of Type E 7 (1)