Investigating relations between discrete Painlevé equations: The multistep approach

International audienceWe show how, starting from a mapping where the independent variable advances one step at a time, one can obtain versions of the mapping corresponding to a multi-step evolution. The same procedure is applied to discrete Painlevé equations, and we proceed to establish Miura relations between the single-step and the multi-step versions (in the present study “multi” referring to double, triple, and quintuple). These Miura relations are discrete Painlevé equations on their own right. We show that, while in some cases it is impossible to obtain a multi-step equation for a single variable, deriving a Miura system is still possible. We perform our analysis for equations associated with the affine Weyl groups E(1)8, E(1)7, E(1)6, and A(1)4