Integrability of $n$-dimensional dynamical systems of type $E_7^{(1)}$ and $E_8^{(1)}$

Details Integrability of $n$-dimensional dynamical systems of type $E_7^{(1)}$ and $E_8^{(1)}$ Integrability of $n$-dimensional dynamical systems of type $E_7^{(1)}$ and $E_8^{(1)}$

2004-09-26

arxiv.org/abs/nlin/0409051v2

Nonlinear Sciences

Exactly Solvable and Integrable Systems

Scientific paper

We propose an $n$-dimensional analogue of elliptic difference Painlev\'e equation. Some Weyl group acts on a family of rational varieties obtained by successive blow-ups at $m$ points in $\mpp^n(\mc)$, and in many cases they include the affine Weyl groups with symmetric Cartan matrices as subgroups. It is shown that the dynamical systems obtained by translations of these affine Weyl groups possess commuting flows and that their degrees grow quadratically. For the $E_7^{(1)}$ and $E_8^{(1)}$ cases, existence of preserved quantities is investigated. The elliptic difference case is also studied.

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