A unified description of the asymmetric q-P_{v} and d-P_{iv} equations and their Schlesinger transformations

Details A unified description of the asymmetric q-P_{v} and d-P_{iv} equations and their Schlesinger transformations A unified description of the asymmetric q-P_{v} and d-P_{iv} equations and their Schlesinger transformations

2003-10-31

arxiv.org/abs/nlin/0310050v1

J. Nonlinear Math. Phys., volume 10, no. 2 (2003) 215-228

Nonlinear Sciences

Exactly Solvable and Integrable Systems

Published by JNMP at http://www.sm.luth.se/math/JNMP/

Scientific paper

We present a geometric description, based on the affine Weyl group E_{6}^{(1)}, of two discrete analogues of the Painlev\'e VI equation, known as the asymmetric q-P_{V} and asymmetric d-P_{IV}. This approach allows us to describe in a unified way the evolution of the mapping along the independent variable and along the various parameters (the latter evolution being the one induced by the Schlesinger transformations). It turns out that both discrete Painlev\'e equations exhibit the property of self-duality: the same equation governs the evolution along any direction in the space of E_{6}^{(1)}.

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