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

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

2010-10-14

arxiv.org/abs/1010.3036v2

SIGMA 7 (2011), 045, 22 pages

Nonlinear Sciences

Exactly Solvable and Integrable Systems

Scientific paper

10.3842/SIGMA.2011.045

We wish to explore a link between the Lax integrability of the $q$-Painlev\'e equations and the symmetries of the $q$-Painlev\'e equations. We shall demonstrate that the connection preserving deformations that give rise to the $q$-Painlev\'e 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 B\"acklund transformation of the $q$-Painlev\'e equation. Each translational B\"acklund transformation may be lifted to the level of the associated linear problem, effectively showing that each translational B\"acklund transformation admits a Lax pair. We will demonstrate this framework for the $q$-Painlev\'e equations up to and including $q$-$\mathrm{P}_{\mathrm{VI}}$.

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