Ralph Willox ; Takafumi Mase ; Alfred Ramani ; Basil Grammaticos - Singularities and growth of higher order discrete equations

ocnmp:13267 - Open Communications in Nonlinear Mathematical Physics, April 16, 2024, Proceedings: OCNMP Conference, Bad Ems (Germany), 23-29 June 2024 - https://doi.org/10.46298/ocnmp.13267
Singularities and growth of higher order discrete equationsArticle

Authors: Ralph Willox ; Takafumi Mase ; Alfred Ramani ; Basil Grammaticos

    We study the link between the degree growth of integrable birational mappings of order higher than two and their singularity structures. The higher order mappings we use in this study are all obtained by coupling mappings that are integrable through spectral methods, typically belonging to the QRT family, to a variety of linearisable ones. We show that by judiciously choosing these linearisable mappings, it is possible to obtain higher order mappings that exhibit the maximal degree growth compatible with integrability, i.e. for which the degree grows as a polynomial of order equal to the order of the mapping. In all the cases we analysed, we found that maximal degree growth was associated with the existence of an unconfining singularity pattern. Several cases with submaximal growth but which still possess unconfining singularity patterns are also presented. In many cases the exact degrees of the iterates of the mappings were obtained by applying a method due to Halburd, based on the preimages of specific values that appear in the singularity patterns of the mapping, but we also present some examples where such a calculation appears to be impossible.


    Volume: Proceedings: OCNMP Conference, Bad Ems (Germany), 23-29 June 2024
    Published on: April 16, 2024
    Accepted on: April 10, 2024
    Submitted on: March 22, 2024
    Keywords: Nonlinear Sciences - Exactly Solvable and Integrable Systems,Mathematical Physics

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