Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 7 (2011), 045, 22 pages      arXiv:1010.3036      http://dx.doi.org/10.3842/SIGMA.2011.045

The Lattice Structure of Connection Preserving Deformations for q-Painlevé Equations I

Christopher M. Ormerod
La Trobe University, Department of Mathematics and Statistics, Bundoora VIC 3086, Australia

Received November 26, 2010, in final form May 03, 2011; Published online May 07, 2011

Abstract
We wish to explore a link between the Lax integrability of the q-Painlevé equations and the symmetries of the q-Painlevé equations. We shall demonstrate that the connection preserving deformations that give rise to the q-Painlevé equations may be thought of as elements of the groups of Schlesinger transformations of their associated linear problems. These groups admit a very natural lattice structure. Each Schlesinger transformation induces a Bäcklund transformation of the q-Painlevé equation. Each translational Bäcklund transformation may be lifted to the level of the associated linear problem, effectively showing that each translational Bäcklund transformation admits a Lax pair. We will demonstrate this framework for the q-Painlevé equations up to and including q-PVI.

Key words: q-Painlevé; Lax pairs; q-Schlesinger transformations; connection; isomonodromy.

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