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

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 t...

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Bibliographic Details
Published in:Symmetry, integrability and geometry, methods and applications integrability and geometry, methods and applications, 2011-01, Vol.7, p.045
Main Author: Ormerod, Christopher M.
Format: Article
Language:English
Subjects:
connection
isomonodromy
Lax pairs
q-Painlevé
q-Schlesinger transformations
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Summary: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-P_{VI}.
ISSN:1815-0659
1815-0659
DOI:10.3842/SIGMA.2011.045