Alfred Ramani ; Basil Grammaticos ; Thamizharasi Tamizhmani - Geometrical description of equations related to the E8 affine Weyl group

ocnmp:11578 - Open Communications in Nonlinear Mathematical Physics, November 13, 2023, Volume 3 -
Geometrical description of equations related to the E8 affine Weyl groupArticle

Authors: Alfred Ramani 1; Basil Grammaticos 1; Thamizharasi Tamizhmani 2

We present a method for the construction of the trajectory of a discrete Painlevé equation associated with the affine Weyl group E$_8^{(1)}$ on the weight lattice of said group. The method is based on the geometrical description of the lattice and the construction of the fundamental Miura relation. To this end we introduce the relation between the nonlinear variables and the corresponding $\tau$ functions. Our approach is heuristic and makes use of some simple rules of thumb in order to derive the result. Once the latter is obtained, verifying that it does indeed correspond to the equation at hand is elementary. We apply our approach to the explicit construction of the trajectory of well-known, E$_8^{(1)}$ associated, discrete Painlevé equations derived in previous works of ours. For each of them we investigate the possibility of defining an evolution by periodically skipping up to four intermediate points in the trajectory and identifying the resulting equation to one previously obtained, whenever the latter exists.

Volume: Volume 3
Published on: November 13, 2023
Accepted on: October 27, 2023
Submitted on: July 12, 2023
Keywords: discrete Painlevé equations,affine Weyl groups,singularity confinement,[NLIN]Nonlinear Sciences [physics]

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