Geodesic
The Lattice Structure of Connection Preserving Deformations for $q$-Painlevé Equations I
Symmetry, integrability and geometry: methods and applications, Tome 7 (2011) Cet article a éte moissonné depuis la source Math-Net.Ru

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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 $q$-$\mathrm{P}_{\mathrm{VI}}$.
Mots-clés : $q$-Painlevé; Lax pairs; $q$-Schlesinger transformations; connection; isomonodromy. 
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     author = {Christopher M. Ormerod},
     title = {The {Lattice} {Structure} of {Connection} {Preserving} {Deformations} for $q${-Painlev\'e} {Equations~I}},
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     url = {http://geodesic.mathdoc.fr/item/SIGMA_2011_7_a44/}
}
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Christopher M. Ormerod. The Lattice Structure of Connection Preserving Deformations for $q$-Painlevé Equations I. Symmetry, integrability and geometry: methods and applications, Tome 7 (2011). http://geodesic.mathdoc.fr/item/SIGMA_2011_7_a44/

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