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

Symmetry Integrability and Geometry-Methods and Applications

article

Christopher M. Ormerod¹ 

[1] La Trobe University, Department of Mathematics and Statistics

关键词: q-Painlev´e;    Lax pairs;    q-Schlesinger transformations;    connection;    isomonodromy;   

DOI  :  10.3842/SIGMA.2011.045

来源: National Academy of Science of Ukraine

【 摘 要 】

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 .

【 授权许可】

Unknown