A systematic method for constructing discrete Painlev\'e equations in the degeneration cascade of the E$_8$ group
Mathematical Physics
2018-01-17 v1 math.MP
Exactly Solvable and Integrable Systems
Abstract
We present a systematic and quite elementary method for constructing discrete Painlev\'e equations in the degeneration cascade for E. Starting from the invariant for the autonomous limit of the E equation one wishes to study, the method relies on choosing simple homographies that will cast this invariant into certain judiciously chosen canonical forms. These new invariants lead to mappings the deautonomisations of which allow us to build up the entire degeneration cascade of the original mapping. We explain the method on three examples, two symmetric mappings and an asymmetric one, and we discuss the link between our results and the known geometric structure of these mappings.
Keywords
Cite
@article{arxiv.1709.06020,
title = {A systematic method for constructing discrete Painlev\'e equations in the degeneration cascade of the E$_8$ group},
author = {Ralph Willox and Alfred Ramani and Basil Grammaticos},
journal= {arXiv preprint arXiv:1709.06020},
year = {2018}
}
Comments
22 pages, 5 figures