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

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 $q$-${\mathrm{P}}_{\mathrm{VI}}$.

References in corpus (4)

• Lax forms of the $q$-Painlevé equations
• A Lax Formalism for the Elliptic Difference Painlevé Equation
• Lax formalism for q-Painleve equations with affine Weyl group symmetry of type E^{(1)}_n
• Galois theory of q-difference equations

Cited by in corpus (1)

• Symmetries in Connection Preserving Deformations